ANNALS OF DISCRETE MATHEMATICS
annals of discrete mathematics Managing Editor Peter L. HAMMER, University of Waterloo, Ont., Canada Advisory Editors C . BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK. OXFORD
ANNALS OF DISCRETE MATHEMATICS
COMBINATORICS 79 PART II
Edited by
M . DEZA, Paris
and I.G. ROSENBERG, Montreal
1980
NORTH-HOLLAND PULILISHING COMPANY
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AMSTERDAM NEW YORK OXFORD
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0 NORTHHOLLAND PUBLISHING COMPANY - 1980
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PRINTED IN THE NETHERLANDS
CONTENTS A. PROSKUROWSKI, Centers of 2-trees
1
E.R. SWART,The edge reconstructibility of planar bidegree graphs
7
U. CELMINS, The Hungarian magic cube puzzle
13
M. MILGRAM, Complete lists of cubic graphs (Abstract)
21
J.-L. FOUQUET, Graphes cubiques d’indice chromatique quatre
23
F. STERBOUL, Un problbme de coloration aux aspects variCs
29
G. SABIDUSSI, Mesures de centralit6 d’un graphe (Abstract)
35
S.A. BURRand P. ERDOS,Generalized Ramsey numbers involving subdivision graphs, and relative problems in graph theory
37
P. FRANKL, A general intersection theorem for finite sets
43
M.O. ALBERTSON, A new paradigm for duality questions (Abstract)
51
R. CORIand J.-G. PENAUD, The complexity of a planar hypermap and that of its dual
53
C. LANDAUER, Acceptable orientations of graphs
63
L. LovAsz, An algebraic upper bound on the independence number of a graph (Abstract)
65
P. ROSENSTIEHL, Preuve alg6brique du critbre de planarit6 du Wu-Lin
67
R.S. WENOCUR, Rediscovery and alternate proof of Gauss’s identity
79
D. BRESSON, Dtcomposition d’un graphe en cycle et chaines
83
B. MONJARDET, ThCorie de la mCdiane dans les treillis distributifs finis et applications
87
P. DUCHET,Graphes noyau-parfaits
93
T. IBARAKI and U. PELED,Threshold numbers and threshold P.L. HAMMER, completions
103
P.L. HAMMER and B. SIMEONE,Quasimonotone Boolean functions and bistellar graphs
107
G.T. KLINCSEK, Minimal triangulations of polygonal domains
121
P.C. KAINEN,Graphs, groups and mandalas (Abstract)
125
C. BERGE,Packing problems (Abstract)
125
V
vi
Contents
J. EDMONDS and W. CUNNINGHAM, Combinatorial decomposition and graphs realizability (Abstract)
126
E.A. BERTRAM, Multipliers of sets in finite fields and b,
127
G. KALAI,Analogues for Sperner and Erdos-KO-Rado theorems for subspaces of linear spaces (abstract)
135
W. WEI, Generalized principle of inclusion and exclusion and its applications
137
P.M. DUCROCQ et F. STERBOUL, Les G-systkmes triples
141
F. STERBOULet D. WERTHEIMER, Comment construire un graphe Pert minimal
147
B. ALSPACH and N. VARMA,Decomposing complete graphs into cycles of length 2p"
155
P. CAMION, Une gCnCralisation dans les p-groupes abCliens CICmentaires, p > 2, des theoremes de H.B. Mann et J.F. Dillon sur les ensembles a differences des 2-groupes abeliens ClCmentaires
163
D. SCHWEIGERT, On correlations of finite Boolean lattices (Abstract)
175
R.P. ANSTEE,Properties of (0, 1)-matrices with forbidden configurations
177
C. VAN NUFFELEN,On adjacency matrices for hypergraphs (Abstract)
181
J.S. BYRNES,Prime triangular matrices of integers (Abstract)
181
B. COURTEAU, G. FOURNIERet R. FOURNIER,Une ghkralisation d'un thkorbme de Goethals-Van Tilborg
183
M. CHEIN and M. HABIB,The jump number of dags and posets: an introduction
189
B. PEROCHE,The path-numbers of some multipartite graphs
195
M. HABIB et B. PEROCHE,A construction method for minimally k-edgeconnected graphs
199
A. AST&-VIDAL,The automorphism group of a matroid
205
P. FLAJOLET, Combinatorial aspects of continued fractions
217
M. POUZET,The asymptotic behavior of a class of counting functions (Abstract)
223
R. MALCOR,The theorem of Whitney and the four colours conjecture (Abstract)
224
J. ZAKS,Non Hamiltonian cubic planar graphs having just two types of faces
225
Contents
vii
A. POLYMERIS, Conjucturally stable coalition structures
229
C. KIRANBABU, Unit distance graphes in rational n-space (Abstract)
235
T.R.S. WALSH,Counting three-connected graphs (Abstract)
235
J.D. MCFALL,Characterizing hypercubes
237
R. CORDOVIL, Sur les orientations acycliques des gComCtries orientees de rang trois
243
and B.D. SAUNDERS, Applications of the Gordon-Stiemke H. SCHNEIDER theorem in combinatorial matrix theory (Abstract)
247
E. MENDELSOHN, Necessary and sufficient conditions that a subgraph of K: can be packed in K: (Abstract)
247
G. SORKIN, The enumeration of nonhomeomorphic graphs by edges
249
C.-F. MA, C.-H. LIUand M.-C. CAI,Optimum restricted base of a matroid
253
R. HAGGKVIST, A characterization of non-Hamiltonian graphs with large degrees (Abstract)
259
J. WOLFMANN, Un problbme d’extremum dans les espaces vectoriels binaires
261
and Z. SKUPIEN, On minimal non-Hamiltonian locally HamilC.M. PAREEK tonian graphs (Abstract)
265
S. RUIZ,On a family of selfcomplementary graphs
267
A. DUCHAMP, Formes bilin’eaires sym’etriques sur un espace vectoriel de dimension finie sur le corps 2 deux ClCments, applications aux matroi‘des 269 binaires V.G. TUPITSYN,On the maximum value of a quadratic form over binary sequences
277
I.S. FILOTTI, Algorithms for determining the genus of a graph and related problems (Abstract)
283
N. ACHUTHAN, On the n-clique chromatic number of complementary graphs
285
On cutting planes A. SCHRIJVER,
291
Problem session
297
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Annals of Discrete Mathematics 9 (1980) 1-5 @ North-Holland Publishing Company
CENTERS OF 2-TREES Andrzej PROSKUROWSKI Department of Computer and Information Science, University of Oregon, Eugene, OR 97403, U.S.A.
The center of a graph G is defined as subgraph of G induced by the set of vertices which have minimal eccentricities (i.e., minimal value of distance to the most distant vertices of G). It has been shown that only a finite number of graphs can be centers of maximal outerplanar graphs (mops). We generalize this result for the class of 2-trees which contains mops.
1. Introduction In a graph G, the distance d(u, v ) between vertices u and u of G is defined as the length of a shortest path connecting u with v (i.e., the number of edges in the path). Eccentricity e ( v ) of a vertex v in graph G is the largest distance from v to any vertex of G. The minimum eccentricity of vertices in G is called the radius r(G). Center C ( G ) of a graph G is the subgraph of G induced by the set of vertices with the smallest eccentricities. In [3] we investigated possible shapes of centers for a class of graphs called maximal outerplanar graphs (mops).
Definition 1.1. A graph is a mop iff it is isomorphic to a triangularization of a polygon. All mops can be constructed according to the following recursive rule (see, for instance, [2]).
Fact 1.2. The “triangle” is the only mop with three vertices. A mop with n vertices can be obtained from some mop M with n - 1 vertices ( n > 3) by adding a new vertex adjacent to two consecutive vertices on the Hamiltonian cycle of M . The main result of [3] follows from two facts: (i) that all centers of mops are nonseparable, and (ii) that there are some forbidden subgraphs for centers of mops.
Lemma 1.3 (see [3]). The center of a mop is nonseparable. Lemma 1.4 (see [3]). Neither of the graphs in Fig. 1 can be a subgraph of the center of any mop. 1
A. Proskurowski
2
Y
Y
Fig. 1. Forbidden subgraphs of centers of mops.
Theorem 1.5 (see [3]). The center of any mop is isomorphic to one of the seven graphs in Fig. 2.
KI
K2
MI
M2
M4
Fig. 2. All centers of mops.
2. Centers of 2-txees We will generalize the above result for so called 2-trees. We define this class of graphs by giving a recursive construction process anologous to that of Fact 1.2 for mops.
Definition 2.1. A 2-tree is a graph that can be obtained in the following recursively defined construction process: (i) The triangle is the only 2-tree with 3 vertices. (ii) Given a 2-tree T with n vertices ( n a 3), add a vertex adjacent to any two adjacent vertices of T. The main difference between mops and general 2-trees is that two adjacent vertices of a 2-tree may have more than two common neighbors. It follows from the recursive construction rule that 2-trees preserve the property of mops that for any two non-adjacent vertices u and v there exist two adjacent vertices whose removal separates u and v.
Lemma 2.2. For any two non-adjacent vertices u and v of a 2-tree T, there exists an edge (x, y ) such that both x and y are adjacent to u and any path from u to v contains x or y .
Centers of 2-frees
3
Proof. We can assume without loss of generality that in a recursive construction of T vertex u has been added according to Definition 2.1, rule (ii) to an edge ( x , y ) of a 2-tree T’ already containing vertex v . According to this rule, n o new vertex can be added to T ‘ U { u }adjacent to both u (or any of u’s descendants in that construction) and to a vertex of T ’ - { x , y}. Thus, the removal of ( x , y ) separates u from v. Inductive use of Lemma 2.2 gives us a statement about the set of edges separating two non-adjacent vertices of a 2-tree.
Lemma 2.3. For any two non-adjacent vertices u and v of a 2-tree T, there are two vertex-disjoint paths connecting u and v and consisting solely of vertices of the separating them edges.
Proof. It suffices to prove the existence of two vertex-disjoint paths from u to end-vertices of an edge (s, t ) separating u from v. The vertices of the paths lie in the same connected component C of T - { s , t } as u and belong to edges separating u and v. Similar paths from v to s and t complete the postulated paths from u to v. We will prove the existence of two vertices x and y in C U {s, t } satisfying the conditions of Lemma 2.2, and with the following property. There exist two vertex-disjoint paths x - s and y - t, possibly degenerated to single vertices, consisting of end-vertices of edge separating u and v. The proof is by induction on the number k of applications of rule (ii), Definition 2.1, necessary to add vertex u in a recurcursive construction of T from a 2-tree already containing vertices v, s, and t. If k = 1, then x = s and y = t. If k > 1 , let us assume that the ( k - 1)st application of rule (ii) involved addition of a vertex u’, adjacent to an edge y’) with the postulated property. As such, u is adjacent to ut and to a vertex z E { x ‘ , y’}, which separate it from v. Taking without loss of generality z = y’, we see that the paths u ’ - x ’ - s and yl- t are vertex disjoint and thus x = u‘ and y = y ’ are the postulated vertices. Edges ( u , x ) and (u, y) extend the above paths to two vertex-disjoint paths from u to s and t. (XI,
The above lemmas allow us to state properties of centers of 2-trees analogous to those of centers of mops.
Lemma 2.4. The center of a 2-tree is nonseparable.
Proof. We will show that for two non-adjacent vertices of the center of a 2-tree T , end-vertices of a separating them edge are also in the center. By Lemma 2.3, this will prove non-separability of the center of a 2-tree. Let us assume that u and v belong to the center of a 2-tree T while a vertex x of a separating them edge ( x , y) does not. This implies that there is a vertex z such that d(x, z)> r(G).If
A. Proskurowski
4
after removal of (x, y ) z is not in the same connected component as u, then e(v)>d(z, u
) l+min(d(y, ~ z ) , d(x, z))amax(d(y, z ) , d(x, z))>d(x, z ) .
This contradicts our assumption that u is in the center. Similarly, if z and the same connected component of T-{x, y}, then e(u) > r(T).
2,
are in
Lemma 2.5. The center of a 2-tree is either K,, K,, or a 2-tree.
Proof. From [l,Theorem 1.13, a graph is a 2-tree if it (i) is connected, (ii) has an edge but not a K4, (iii) has edges for all minimal separating subgraphs. A nonseparable induced subgraph S of a 2-tree T satisfies clearly (i) and (ii), and has no single separating vertex. If there existed two non-adjacent vertices separating S, then S, and thus also T, would have an induced cycle of length greater than 3, prohibited for 2-trees. A separating set of more than two vertices implies existence of more than two vertex disjoint paths between two vertices in T, a contradiction with [l, Theorem 3.51. Thus (iii) is satisfied by S, and S is a 2-tree. The separation property expressed by Lemma 2.2 allows a generalization of Lemma 1.4.
Lemma 2.6. Neither of the graphs in Fig. 1 can be a subgraph of the center of any 2-tree.
Proof. Let us assume that a graph G (G, or Gb from Fig. 1) is an induced subgraph of the center of a 2-tree T. Clearly, the edge (x, y ) separates w' from w" in T. Let us assume that a vertex z such that d(y, z ) = r(T) is not in the same connected component of T-{x, y} as w, WE{W',w"}. Then e ( w ) > d ( w , z ) > d(y, 2) = r(T) contradicting the assumption that w is in the center of T. We will now describe all 2-trees which do not have graphs of Lemma 2.6 as subgraphs. To facilitate the description we will use the following notion of a set of vertices commonly adjacent to end-vertices of an edge.
Definition 2.7. For a given edge (x, y ) of a 2-tree T, we define a 2-star to be the set of all vertices of degree 2 of T adjacent to both x and y. A schematic representation of a family of 2-trees parametrized by the size of a 2-star is given in Fig. 3.
, Fig. 3. A family of 2-trees.
, ... etc.
Centers of 2-trees
5
Let us consider outerplanar subgraphs of such a family. We readily observe that almost all 2-trees in the family have the same set of mop subgraphs, as 2-stars of increased sizes do not introduce any new outerplanar subgraphs (the only exception are the 2-trees with 3 and 4 vertices which belong to the same family). Therefore, any graph which can be the center of a mop may give origin to families of 2-trees which avoid the forbidden subgraphs of Lemma 2.6 as well. These families are obtained by substituting 2-stars for all vertices of degree 2 in the mops. Only one of the mops in Fig. 2 has an internal edge (not on the Hamiltonian cycle) with end-vertices not adjacent to a vertex of degree 2 (viz., (x, y ) of M.5). Appending a 2-star to this edge does not introduce either of the forbidden subgraphs. By inspection we find that the addition of vertices adjacent to the introduced 2-stars either results in a 2-tree of another family of this group, or introduces a forbidden subgraph of Lemma 2.6. This implies our main theorem.
Theorem 2.8. The center of any 2-tree is isomorphic to a member of the families of 2-trees in Fig. 4.
Fig. 4. Families of centers of 2-trees.
3. summary We have given a simple characterization of the graphs which can be centers of 2-trees. These graphs belong to families of 2-trees obtained by parametrization of the admissible centers of maximal outerplanar graphs. We proved our results using the fact that 2-trees are separable into bi-connected components by removal of any internal edge. This implies the existence of few configurations which are forbidden in the center of any 2-tree.
References [l] D.J. Rose, On simple characterizations of k-trees, Discrete Math. 7 (1974) 317-322. [2] A. Proskurowski, Minimum dominating cycles in 2-trees, Int. J. Comput. Information Sci. 8 (5) (1979) 405-417. [3] A. Proskurowski, Centers of maximal outerplanar graphs, J. Graph Theory 4 (2) (1980) 75-79.
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Annals of Discrete Mathematics 9 (1980) 7-12 @ North-Holland Publishing Company
THE EDGE RECONSTRUCTIBILITY OF PLANAR BIDEGREE GRAPHS E.R. SWART Uniuersity of Waterloo, Waterloo, Ont., Canada
0. Introduction The edge-reconstruction conjecture is concerned with the question of whether or not a graph can be reconstructed from its edge-deleted subgraphs. For any graph G with m = lEGl edges we can imagine setting up a deck of m cards each of which contains a different edge-deleted subgraph of G, formed by the deletion of a single edge. A graph G is then said to be edge-reconstructible if all possible reconstructions of G from these m cards are isomorphic to G. It has been suggested by Harary [4] that all graphs with four or more edges are in fact edge-reconstructible as set out in the following conjecture.
1. Edge-reconstruction Conjecture All finite strict undirected graphs with at least four edges are edgereconstructible. This conjecture is known to be true for disconnected graphs (with at least two non trivial components), trees, regular graphs and certain other special classes of graphs. The conjecture is somewhat weaker than the original vertexreconstruction conjecture and it can be shown that every graph, without isolated vertices, which is vertex reconstructible is also edge-reconstructible [3]. It follows in particular that separable graphs without pendant vertices are edgereconstructible [2]. It is not, however, known to be true for planar graphs or for bidegree graphs [3] and the present paper is concerned with the characterisation of a potential bidegree counterexample to the edge-reconstruction conjecture. As it turns out the nature of the constraints on a bidegree counterexample make it possible to establish the truth of the following theorem.
Theorem 1. All finite stricf bidegree graphs with four or more edges which are embeddible in surfaces with Euler characteristic x 2 - 1 are edge-reconstructible. It follows that, in particular, all planar bidegree graphs are edge-reconstructible and in order to establish this result it is necessary to build up a set of excludable 7
E . R . Swart
8
configurations which cannot occur in any minimum bidegree counterexample, G say, to the edge-reconstruction conjecture. If the degrees of the two classes of vertices in G are 6 and A respectively with A > 6 , then it is not too difficult to show that (1) G is not a disconnected graph; (2) G is a graph with minimum degree 3 2 ; (3) The degrees of the vertices in G are such that A = 6 + 1 : (4) G contains at least two vertices of degree 6; ( 5 ) G does not have a cut edge. In extending these results so as to create a set of excludable configurations we can make use of the fact that the degree sequence of a graph is edgereconstructible [3] as well as the fact that the number of k-gons for k 2 3 in any strict graph with four or more edges is edge-reconstructible. This latter result is really just a special case of Kelly's original lemma [6] on the number of subgraphs of specified type, for the vertex deletion case as transposed to the edge-deletion case [3]. Two simple results will serve to illustrate the techniques employed: If we denote the vertices of degree 6 by a 0 and the vertices of degree A by a A, then the configuration
A is excludable since if the edge (Y is deleted this creates two vertices of degree 6 - I and in order to recover the correct degree sequence the missing edge (Y can only be replaced in its original position. The excludability of the configuration
cannot be established in such a simple manner but it can be deduced from the chain of reasoning shown below where Gidenotes a graph isomorphic to G and Hia graph which is not isomorphic to G but is isomorphic to all other graphs Hi and has the same edge deleted subgraph as G.
G2
i)G,-6
Fig. I .
H2
A
The edge reconsfructibilify of planar bidegree graphs
9
Since H 2 = G , it is clear that Hi= G, (isomorphic) for all Hi and all G,, which is a contradiction. Among the induced subgraphs or configurations which can be shown to be excludable by various extensions of the above type of argument the 24 subgraphs shown in Fig. 2 are of interest for the present discussion [ 8 ] .
3
0 Fig. 2 .
E.R. Swart
10
A subset of this set of excludable configurations has been obtained by Hoffman [5] using a similar line of argument. Now if we consider a “central” small vertex of degree 6, then, in the light of these excludable configurations, it follows that it can only have two possible environments up to 3rd neighbour level as illustrated in Fig. 3 for the case A = 3 and 6 =
Case 1
Case 2
Fig. 3.
In both cases all the vertices shown are distinct. We know, moreover, from our set of excludable configurations that a large vertex can never be a second neighbour to more than two small vertices, and in addition, the vertex u in Case 1 cannot be a second neighbour to a small vertex other than the central vertex itself. It follows that for Case 1 n1 1 2 -s A 1 +;((A 1 ) 2 ( A I ) ) + A 2 A 2 + A -4 n2 and for Case 2 n1 1 n2 ( A - l ) + i ( A - 1 ) 2
-s
-
2 A’-1’
Thus overall we can say that if n, is the number of vertices of degree 6 and n2 is the number of vertices of degree A = 6 + 1 -n1s - 2 n2 A ’ - 1 and this allows us to set up Table 1 where Table 1 A 3
4 5 6
k=(n,/n,)max
pmin
I
14
2 -
66 17
12 2 35
220
4
15 1 -
5
64
13
37
p
is the mean vertex degree in G
11
The edge reconstrucfibility of planar bidegree graphs
Now for any graph embedded in any surface with Euler characteristic
x
lFGl+ 1 VGJ= lEGl+ x or
f+n=m+X. Moreover
pn=2m where
p*
and
P*f=2m
is the mean degree of the faces in the embedding in question. Thus
( p2 + ; 2) m = m + x . And since we know from our table that p a y and, from the list of excludable configurations, that the girth of our graph G 2 8 we get
2 2x5 -+--l)mzX 8 14
or
1 ---ax. 28
It follows immediately that the Euler characteristic of G must be negative for all possible choices of A. This implies that the smallest possible counterexample would have to be a subdivision of the (3,7) cage which has 24 vertices [ 7 ] . And in view of characteristic (4) ($2 small vertices of degree 6) this means that our minimum counterexample would have to have $26 vertices. The corresponding minimum number of edges would have to be 38 but this gives
which implies that x s - 2 and this completes the proof of Theorem 1. The following two theorems can also be established.
Theorem 2. Every bidegree line graph with four or more edges is edgereconstructible. Proof. Our hypothetical counterexample G always contains K,.3 as an induced subgraph which is one of the forbidden configurations for line graphs [l]. Theorem 3. Every bidegree bipartite graph with four or more edges is edgereconstructible. Proof. The proof of this latter theorem depends on the fact that bipartite graphs are recognizable [ 2 ] and the details are left as an interesting exercise for the reader. References [ I ] L.W. Beineke, Derived graphs and diagraphs, in: H. Sachs, H. Voss and H. Walther, Eds., Beitrager zur Graphentheorie (Leubner, Leipzig. 1968) 17-33. [2] J.A. Bondy, On Ulam’s conjecture for separable graphs, Pacific J. Math. 31 (1969) 281-288.
12
E . R . Swart
[3] J.A. Bondy and R.L. Hemminger, Graph reconstruction - A survey, J . Graph Theory 1 (1977) 227-268. [4] F. Harary, On the reconstruction of a graph from a collection of subgraphs, in: M. Fielder, Ed., Theory of Graphs and Its Applications (Proceedings of the symposium held in Prague) (Czechoslovak Academy of Sciences, Prague, 1964) 47-52 (reprinted, Academic Press, New York). [ 5 ] D.G. Hoffman, Notes on edge reconstruction of bidegree graphs, privately circulated. [6] P.J. Kelly, On isometric transformations, Ph.D. Thesis, University of Wisconsin (1942). [7] W.F. McGee, A minimal cubic graph of girth seven, Can. Math. Bull. 3 (1960) 149-152. [ 8 ] E.R. Swart, The edge-reconstructibility of planar bidegree graphs, University of Waterloo Research Report CORR 78-44.
Annals of Discrete Mathematics 9 (1980) 13-20 @ North-Holland Publishing Company
THE HUNGARIAN MAGIC CUBE PUZZLE Uldis CELMINS Department of Combinatorics and Ont. N2L 3G1, Canada
Optimization, University of
Waterloo,
Waterloo,
The puzzle is a mechanical device seemingly consisting of 27 little cubes arranged in a large 3 ~ 3 x cube. 3 In its initial configuration each face of the large cube is coloured a different colour. Thus each little cube has 0, I , 2 or 3 of its faces coloured. An arbitrary configuration is obtained by breaking up the large cube and reassembling it so that each little coloured face remains visible. We may also picture the large cube in three different ways as 3 layers of 9 cubes each. The device is constructed to allow face-turns only. by which is meant a 90" rotation of an outside layer with respect to the other two layers. An accessible configuration is one obtainable by a sequence of face-turns. We demonstrate (a) a method for deciding whether an arbitrary configuration is accessible or not (there are 12 orbits or classes of configurations). (b) a procedure for obtaining the initial configuration from any accessible one by face-turns. (Current best is at most 100 turns.) We generalize the above to certain cubic maps.
1. introduction and definitions Consider a cubic map M obtained by properly embedding a graph in an orientable surface. The boundary of each region is a simple circuit and thus gives a face of M. In the case of the puzzle the underlying map is the planar embedding of the cube. By colouring the faces we induce a labelling of the edges and vertices of M. These induced labels become the visible colours of the edge-cubes and vertex-cubes respectively. To work with the puzzle on paper we move edge-cubes from edge to edge and vertex-cubes from vertex to vertex on the map. A face-turn generates a particular cyclic permutation of the edge-cubes (resp. vertex-cubes) on that face. In the initial configuration we say each edge-cube and vertex-cube is fixed in position. After several face-turns a cube may return to its position o n M but might no longer be fixed. An edge-cube might be flipped and a vertex-cube might be twirled clockwise or counterclockwise. Thus, if M has v vertices and e edges the number of arbitrary configurations is u ! e ! 3"2'. A turn diagram (see Fig. 1) consists of the map M and a collection of sequentially numbered arrows placed alongside edges in the faces of M. Each arrow is meant to indicate a particular face-turn. We show the result of performing the indicated sequence of turns, called a move, by a vertex-cube permutation diagram and an edge-cube permutation diagram. For each cube that is displaced 13
.
14
U. Celttiins
an arc is formed by piecing together the arrows of the turn diagram. These arcs are then simplified to display in a concise manner the effect of the move. The resulting permutation diagram may therefore not contain enough information to invert the move. Since it is easy to draw a permutation diagram for any particular configuration, the central problem in obtaining an initial configuration from an accessible one is this loss of information. These permutation diagrams reveal the cycle structure and hence parity of the edge and vertex permutations generated by a move. In addition each such cycle is said to have a nature. If we take a k-cycle in the edge (resp. vertex)-cube permutation and perform the move that creates this cycle k times, then each edge-cube (resp. vertex-cube) returns to its original position. However. all edge cubes will be flipped or none will be flipped (resp. all vertex-cubes will be twirled clockwise, or all twirled counterclockwise. or none twirled at all). Hence an edge-cube (resp. vertex-cube) cycle is named a flip-cycle or a non-flip-cycle (resp. clock-twirl-cycle, counter-twirl-cycle, non-twirl cycle). We speak of the natured cycle structure of a configuration. We say an edge-cube permutation is non-Pip according as t h e number of its flip-cycles is even or odd. We say a vertex-cube permutation is non-twirl, clocktwirl, or counter-twirl according as the number of its clock-twirl cycles minus the number of its counter-twirl cycles is congruent to 0, 1 or 2 (mod 3).
2. The visual rule and a theorem on accessible configurations
In this section we present results which are concerned with both the edge-cube and vertex-cube permutations. Since the development of the theory is similar, we may sometimes concentrate on the vertex-cube permutations. To each cycle there corresponds a closed directed curve on the surface containing the map. For edge-cube permutations the direction is irrelevant and is dropped. The curve is obtained by piecing together the arcs of the permutation diagrams as in Fig. 1. Notice that the nature of the cycle can be determined from the curve and is independent of the number of vertex-cubes or edge-cubes permuted by that cycle. Consider a closed curve corresponding to a cycle in a vertex-cube permutation. In what follows, we assume the curve separates the surface into two regions, “inside” and “outside”. We adopt the convention that the curve is directed clockwise around the inside region. Suppose we further assume that the curve does not intersect itself and that t h e inside region is but a single face F. If an edge bounds F but is outside the curve. that indicates a turn of face F. A vertex-cube which is displaced by this face-turn has a face in F and as indicated in Fig. 2 this little face remains in F. If an edge bounds F but is inside the curve, that indicates a turn of a face F‘ different from F. The vertex-cube which is displaced by this face-turn has a face in F and as
15
The Hungarian Magic Cube Puzzle
piJ 3
Turn diagram for primitive B.
Vertex-cube permutation diagrams, initial and simplified.
The arc from vertex x to vertex y indicates that the vertex-cube that is displaced from x to y moves as if it has been turned clockwise by face 6 and then clockwise by face 4.
Curve containing no vertices. Nonflip.
Edge-cube permutation diagrams, initial and simplified.
Directed curves. C , is clock-twirl and C , is countertwirl.
Permutation diagrams for (primitive €3)’. The four displaced vertex cubes return to their positions. but two are twirled clockwise and two counter-clockwise as indicated. We use the “wiggly” arc in the edge-cube permutation diagram as a shorthand for a displacement with flip.
Permutation diagrams for (primitive B ) 3 . The edge-cubes are now fixed in position. We use the “wiggly” arc in the vertex-cube permutation diagram to denote the only displacement between adjacent vertices that cannot be given by a single face-turn.
Fig. 1. Explanation of notation.
non-twirl cycle
counter-twirl cycle
clock-twirl cycle
Fig. 2. The visual rule.
non-twirl cycle
16
U . Celmins
indicated in Fig. 2, this little face turns away from F in a counter-clockwise direction. From here we proceed by induction on the number of faces in the inside region to obtain the following visual rule:
Lemma 1. I f the curve is non-self-intersecting and if it separates the surface into two regions and is directed clockwise around the inside region, the cycle in the uertex-cube permutation is non-twirl, counter-twirl, or clock-twirl according as the number of edges inside the curve is congruent to 0, 1 or 2 (mod 3). We also have: A cycle in the edge-cube permutation is a flip-cycle i f fthe number of vertices of the map inside the curue is odd.
P and P are non-twirl P'
P
vertex-cube permutation diagrams
arc-decomposition
vertex-cube permutation diagrams
arc-decomposition
Q and Q' are coun ter-twirl
PO and P'Q' are counter-twirl
@ PQ
lg/ P'Q'
vertex-cube permutation diagrams
arc-decomposi tion A
Simplified vertexcube permutation diagrams for PQ and P'Q' PQ
Fig. 3. Lemma illustrations.
The Hungarian Magic Cube Puzzle
17
The arc-decomposition (see Fig. 3 ) of a vertex-cube (resp. edge-cube) permutation is obtained by taking each arc alongside k 2 1 edges and drawing it as k arcs, each alongside one edge. It is possible to apply the visual rule to self-intersecting curves (see Fig. 3, Q and Q’) because of the following lemma:
Lemma 2. Two vertex-cube (resp. edge-cube) permutations with the same arcdecomposition have the same nature. In addition we have these lemmas: Lemma 3. I f move P has the same arc-decomposition of both the vertex-cube and edge-cube permutations as move P’ and if moves Q and Q’ are also related in this way, then the product moves PQ and P’Q’ can also be so related. (PQ denotes move P followed by move Q, see Fig. 3 ) . Lemma 4. If the vertex-cube (resp. edge-cube) permutation of moves P and Q contains just one cycle of k > 1 vertex-cubes (resp. edge-cubes) and these two cycles (one from P and one from Q ) have no vertex-cubes (resp. edge-cubes) in common, then the nature of the vertex-cube (resp. edge-cube) permutation of PQ is the “sum” of the respective natures of P and Q. Lemmas 2, 3 and 4 are used to prove the following lemma:
Lemma 5. Given any two moves P and Q, the nature of the vertex-cube (resp. edge-cube) permutation of PQ i s the “sum” of the natures of the vertex-cube (resp. edge-cube) permutations of P and Q. Given a Hungarian Cube Puzzle in an arbitrary configuration (or a general map puzzle) we first draw the permutation diagrams that would yield the initial configuration. We can then determine if this configuration is accessible or not by the following theorem:
Theorem 6. For any map M and initial face colouring, an arbitrary configuration is accessible only i f (i) the parities of the edge-cube and vertex-cube permutations are the same, (ii) the edge-cube permutation is non-flip, (iii) the vertex-cube permutation is non-twirl. Part (i) follows from t h e fact that each face-turn yields a k-cycle in both the vertex-cube and edge-cube permutations. Parts (ii) and (iii) are a consequence of Lemma 5 and the fact that each
18 turn diagrams
vertex-cube permutation diagrams
edge-cube permutation diagrams
Basic vertex-cube permutation moves A and B. Move A followed by the inverse of B causes two adjacent vertex-cubes to he twirled: one clockwise, the other counter-clockwise.
A:
)=%()-( -1- 3-
) k %
Primitive moves A and B. To solve the general orientable surface map puzzle we use the primitive moves first to fix the edge-cubes in position and then the basic vertex-cube moves to fix the vertex-cubes. Adjacent edge-cubes can be flipped by following B with the mirror-image in the indicated axis. The number of turns required is at most a quadratic polynomial in the number of vertices. Fig. 4. General moves for arbitrary maps
face-turn yields a non-twirl cycle of the vertex-cube permutation and a non-flip cycle of the edge-cube permutation. The theorem partitions the arbitrary configurations into 12 classes (or 24 if each face is bounded by an odd number of edges). The moves given in Fig. 4 show the converse of the theorem to be true provided M is of girth at least 4 and provided no two faces have more than one edge in common. (This last condition excludes, for example, the three toroidal embeddings of the cube.)
19
The Hungarian Magic Cube Puzzle
3. Solving the cube puzzle We conclude with some remarks on obtaining the initial configuration from any accessible one for the Hungarian Magic Cube Puzzle, that is to say, on solving the cube. In giving demonstrations, we use a layer by layer approach that requires the memorization of five moves and needs no consultation of tables. This form may take as many as 1.50 face-turns although 100 is often the case. When tables are consulted the total number of face-turns can always be reduced to about 100. First, the edge-cubes and then the vertex-cubes of the first layer are fixed in position (this exercise is left to the reader). Then, the two moves in Fig. 6 are used to fix the edge-cubes of the middle and last layers. Finally the basic vertex-cube permutation moves A and B of Fig. 4 are used to fix the vertex-cubes of the last layer. The fifth move to memorize is the use of A and B together to twirl vertex-cubes, one cube clockwise and one counter-clockwise. It seems to be desirable to have short moves that produce small changes in the cube. This author knows three &turn moves that give a 3-cycle of vertex-cubes only, permuting no edge-cubes at all. A and B are given in Fig. 4 and C is
Example 1. A 2-nesting of the basic vertexcube permutation move B. (i) turn faces I , then 2 counterclockwise. (ii) Apply move B where indicated (see Fig. 4). (iii) Turn faces 2. then 1 clockwise. The resulting 12 turn move is given at the right. The edge-cubes remain fixed in position.
A 2-nesting of the primitive B obtained by first turning face 1 clockwise and face 2 counter-clockwise
The primitive B in another position.
(first turn face
3 clockwise)
The product move of the above two moves (12 face-turns). We use “ X ’’ to indicate flips of edge-cubes.
A I-nesting of the product move (14
Fig. 5. Nesting of moves.
turns)
U. C h i n s
20 \
\
Use this move and its mirrorimage in the indicated axis to fix edge-cubes of the middle layer. I
I
\\
Use this 1-nesting of the primitive A to fix the edge-cubes of the last layer in position. This move with its mirror-image in the indicated axis will flip adjacent edge-cubes.
Fig. 6 . Some other moves for the cube puzzle.
obtained by reversing the directions of face-turns 3 and 7 in move B. We also have two %turn moves that permute three edge-cubes only and a &turn move that gives a 5-cycle of edge-cubes only. Of the preceding moves the ones with 8 turns we term basic. In addition we make use of two primitive moves of four face-turns each (Fig. 4). both of which permute 3 edge-cubes and two pairs of vertex-cubes. A k-nesting of a move X is a new move Y , consisting of k face-turns followed by the move X , followed by the inverse of the k face-turns. It is not hard to show that the natured cycle structures of X and Y are the same. We thus say that moves X and Y are similar. We find that all 15 non-twirl 3-cycles of vertex-cubes can be obtained by a k-nesting of move C where k is at most 4. All 27 non-flip 3-cycles of edge-cubes can be obtained from another basic move by k-nesting with k at most 4. The author has tabulated these and the 1- and 2-nestings of the primitive moves. Fig. 5 contains some examples. This material suggests two algorithms for solving the cube: use the primitive moves or otherwise fix all the edge-cubes (resp. vertex-cubes) in position and then use the basic vertex-cube permutation moves (resp. edge-cube) and their k nestings to fix the remaining cubes in position. The number of face-turns required can be reduced to less than 100 by careful use of tables. We conjecture that all configurations similar to a given one can be obtained by a k-nesting of it, where k is small, perhaps at most 10. This might lead to the following procedure: (i) given an accessible configuration, make several turns to obtain another configuration similar to a catalogued one. (ii) compute the k-nesting of the catalogued configuration that is needed to solve the cube. Since presenting an earlier version of this paper in Montreal in June 1979, I have heard of several people who have worked on the puzzle, among them David Singmaster and John Conway in England, Ervin Bajmoczy in Hungary and Don Taylor in Australia. I wish to thank Frank Allaire for giving me my first Hungarian cube and for the many ensuing hours of enjoyment.
Annals of Discrete Mathematics 9 (1980) 21 @ North-Holland Publishing Company.
COMPLETE LISTS OF CUBIC GRAPHS Martin MILGRAM Silver Spring, Maryland, USA
Abstract In [l] we outlined a general plan for listing all graphs irreducibly nonrepresentable on a surface S. We started with the observation that each representation problem could be reduced to one involving cubic graphs. First we enumerate all Kagno graphs of types I and I1 [2]. Then we list all graphs where exactly three edges disconnect using the method of [3]. The remaining graphs have forbidden configurations. In [l] we illustrated this technique first by giving a very simple proof of the finiteness of irreducibly non-planar graphs and then the finiteness of graphs non-representable on the projective plane. [3] gives the complete list of graphs on the projective plane using techniques directly analogous to Kuratowski’s classic result. The torus is clearly the next surface to consider. Kagno knew of ten cubic graphs of types I and 11 which are 2-irreducible. The techniques of [4] gave an additional six graphs which complete the list of 3-edge connected 2-irreducible cubic graphs. This, with some forbidden configurations, is sufficient to show the finiteness of a 2-irreducible cubic graph. However, to obtain the complete list, we need an auxiliary list: all 14 node non-planar cubic graphs such that n o three edges disconnect. The bounds in [5] make tractable the implementation on a microcomputer of an algorithm which builds complete lists of such cubic graphs, demonstrates nonisomorphism in pairs and gives all their representations on a torus.
References [ 11 M. Milgram, J. Combinatorial Theory 12 (1972) 6-31. [2] I.N. Kagno, J. Math. and Phys. 16 (1973) 46-75. [3] M. Milgram, J. Combinatorial Theory 14. [4]M. Milgram, Isreal J. Math. 19 (3) (1974) 201-207. [5] M. Milgram and P. Ungar, J. Combinatorial Theory 23 (1977) 227-233.
21
This Page Intentionally Left Blank
Annals of Discrete Mathematics 9 (1980) 23-28 @ North-Holland Publishing Company
GRAPHES CUBIQUES D’INDICE CHROMATIQUE QUATRE Jean-Luc FOUQUET 187 rue D’lsaac, 7200 Le Mans, France In this paper we give some properties of cubic graphs whose edges are not colorable with three colors. We give then a method to show that there is no “snark” with exactly 16 vertices. At the end, we give a new construction of snarks.
1. Introduction (1) DCsignons par 9 I’ensemble des graphes cubiques connexes, d’indice chromatique 3 (les arCtes de G E sont ~ coloriables en 3 couleurs), par 9 I’ensemble des graphes cubiques, connexes, d’indice chromatique 4: (2) Pour G (G = ( X , E ) )E 9 U22 OG {@ 1 @ : E + ( a P y 8 ) ) est l’ensemble des colorations des arCtes de G (V@ E OG Ve, e ’ E E adjacentes @ ( e )# @(e’)); E d w ) = { e I e E E, W
e )= w ,
V w €{a,P, Y , 8))
par convention nous supposerons que IE,(8)1 S lE,(a)l si G E ~V@EOG , E,(6)=4. Nous dirons que @ E OG est propre (@ E 9)si v[a, b ]E E
@[a,b l = 6 3 VO { a ,P, r>
S
&(w)
avec E ,
IE,(p)I
G J E , ( y ) (et
que
n Euf 4
= {[x,
al[y, a], [ z , bl, [f, bll.
x et y Ctant les voisins de a (autres que b ) dans G, z et f ceux de b (autres que a ) . Nous dirons que @ E Oc est forte (@ E 9)si & ( 8 ) constitue un sous graphe de
G (il n’y a pas d’arCtes de G incidentes a deux arCtes de E , ( 6 ) ) . Nous dirons enfin que @ est 6-minimum si @ a exactement s ( G ) = min4,,,G IE,(6)) arCtes dans E,(6) (@ &minimum 3 @ E 9). (3) Soit G E et~@ ~ 9Les . ariites de E,(6) sont de trois sortes, respectivement A,, B, et C,.
B b
23
24
J.-L. Fouquef
2. Proprietes des graphes de 9
Theoreme 1. Soit G = ( X , E ) E 22 alors V @E 9 on a IA,l = 1B,J = IC,l= 1E,(6)1
(mod 2)
Preuve. (a) Supposons [A,I = 1 (mod 2), IB,I = IC,l = O (mod 2) chaque ar&tede A, et B, porte un (et un .seul) sommet de degrC 1 dans @(a,p ) (graphe partiel de G muni de @, engendrC par les deux couleurs a et p). Chaque arCte de C, porte deux sommets de degrC 1 dans @(a,p ) . Tous les autres sommets de G seront de degrC 2 dans @(a,p ) et donc @(a,p ) a lA,I+ IB,1+2 IC,l sommets de degrC 1 avec (A,I + IB,\ + 2 IC,l= 1 (mod 2) ce qui est impossible. (b) IA,I=l (mod2), IB,I=l (mod2). IC,l-O (mod2); il suffit d'examiner @(p, y) et nous obtenons le m&me genre de contradiction. Corollaire 1.1. Soit G = ( X , E ) E 9 alors V @ E 9 IE,(S)lz 2, en particulier s(G)3 2. Proposition 2. Soit G = ( X ,E ) E 9, @ E OG 8-minimum, e E &(a)+ il existe un cycle e'le'mentaire de longueur impaire contenant e (note' c,) dont toutes les ar2tes sont colorie'es par 2 couleurs seulement (parmi a p y ) sauf e. Preuve. Soit e = [x, y ] E A, alors la chaine CICmentaire de @(p,y) d'origine x a pour extrCmitC y ; sinon en permutant les couleurs p et y sur la chaine contenant x on obtiendrait une nouvelle coloration @'# 9 de laquelle on peut dtduire @"E 9 avec IE,.(S)l < IE,(6)) (on pourrait alors recolorier e en p ou y). Le cycle C, cherchC est donc constituC de cette chaine et de e. Corollaire 2.1. Soit G = ( X , E ) E ~@ , € a 6-minimum: , soient el = [ x l ,yl]eA,, e2 = [x,, y 2 ] € B,, e3= [x,, y , ] ~C,; alors le sous graphe de G engendre' par {x,, x2, x,, y l , y 2 , y,} ne contient pas d'autres arites que e,, e2 et e3. Theoreme 3. Soit G = ( X , E ) E 9 @ 6-minimum E aG,el et e2 deux ar2tes quelconques de E,(6) alors:
De plus se el et e2 appartiennent h deux classes distinctes alors V x E C,, V y E Ce2 d ( x , y ) 2 2. (Ce rCsultat Ctant le meilleur possible.)
Corollaire 3.1. Soit G = ( X , E ) €9 alors G possbde au moins s ( G ) cycles e'le'mentaires de longueurs impaires disjoints deux h deux.
Graphes cubiques d’indice chrornatique quafre
25
Les preuves de ces corollaires et ce thCrkme sont trop longues pour pouvoir ttre exposies ici. Payan dans [ 5 ] a montrt que quelque soit G = ( X , E ) E 9, G posskde une coloration forte. En fait les propriCtC prkctdentes nous permettent de dCmontrer le
Theoreme 4. Soit G = ( X , E ) E 22 alors il existe @ 8-minimum et 4 E 9. 3. I1 n’existe pas de snark ayant exactement 16 sommets
Un ‘ h a r k ” est un graphe de 2 sans triangle ni carrC, 3 arkte-connexe, cycliquement 4 artte-connexe (l’enlkvement de 3 arktes quelconques ne disconnecte pas le graphe en deux sous-graphes, chacun d’eux contenant un cycle) voir [3]. Le snark Ctait alors considCrC comme Ctant un “animal” mystCrieux et rare, la chasse aux snark prit un Clan nouveau griice a I’article de Isaacs [4] en 1975 qui dCcouvrit une famille infinie de “marks”. Le premier snark connu est naturellement le graphe de Petersen, en fait V n 2 18 nous connaissons au moins un snark d’ordre n ; comme il n’existe pas de snark a I’ordre 12 ni 14, on s’est longtemps demand6 s’il pouvait en exister 2 I’ordre 16: ceci est I’objet de cette partie. (Nous ne donnerons ici que le canevas de la dkmonstration.) Nous Ctablissons successivement les deux lemmes suivants: Lemma 5.1. Soit G = ( X , E ) un mark ayant exactement 16 sommefs, alors il ne posstde pas de sous-gruphe isomorphe a H (Fig. 1).
Fig. 1.
Preuve. La preuve est simple, il suffit en effet de compter le nombre d’arttes possibles dans G X - H , puis d’inventorier les sous-graphes possibles pour G X - H(il y en a 5 ) : chaque cas nous menant a un graphe 3-coloriable: contradiction. Lemme 5.2. Soit G = ( X , E ) un mark ayant exuctemenf 16 sommets alors il ne possbde pas de sous-graphe isomorphe a K (Fig. 2).
Fig. 2.
26
J.-L. Fouquet
Preuve. M$me mCthode que le Lemme 5.1. Dks lors, d’aprks le thiorkme de Petersen, si G est un snark d’ordre 16, il admet I’un des trois 2-facteur suivants: cycles de longueur 5 et un cycle de longueur 6 , 5 et un cycle de longueur 11, - 1 cycle de longueur 7 et un cycle de longueur 9.
-2
- 1 cycle de longueur
Comme G ne peut alors contenir ni H, ni K, il est facile alors de virifier qu’il n’existe pas de snark ayant 16 sommets. Remarquons enfin que cette meme technique permet de montrer qu’il n’y a pas de snark ayant 14 sommets sans utiliser le listing des graphes cubiques d’ordrea 14 Ctabli dans [l].
4. Construction de snarks L’objet de cette partie est de montrer que I’on peut construire autant de “snark” cycliquement 4-arCtes connexes que I’on disire (voir [2] pour plus de prCcisions sur ce sujet). Soient G = ( X , E ) E 9 et G’ = ( X ’ ,E’)E 9
X
=X’
[x, y ] E[z, ~ t ] E E (et [x, t ] f! E[y, z] $ E).
Les arktes de G’ sont constituCes des arctes de G a l’exception des arEtes [x, y] et [z, t ] remplacies par [x, t] et [y, 21
Exemple. X
G
z
X
G‘
Soit alors H le graphe obtenu a partir de G (et G’) de la manikre suivante: supprimons les ar&tes[ z , t ] et [x, y] de G et ajoutons deux nouveaux sommets a et b reliis par une arCte entre eux et aux sommets x et z pour a, y et t pour b. On a alors la
Proposition 6. Si
alors HE9.
27
Graphes cubiques d’indice chromatique quatre
La dtmonstration de cette proposition est triviale, mais ceci nous permet de construire des “snarks”. I1 suffit de remarquer que dans le graphe de la Fig. 3 ci dessous, les ar&tes[xl, all et [x,, a,] sont toujours colorites de la mCme couleur (ainsi que [x,, a,] et [x,, a,]; sinon le graphe de Petersen de la Fig. 3‘ serait 3 coloriable ou n’admettrait qu’une seule arCte de couleur 6, ce qui est impossible.
Fig. 3.
Fig. 3’
Construisons alors une chaine de graphes comme celui de la Fig. 3. Complttons alors le graphe de la Fig. 4 de manibre 9 le rendre cubique soit G ce graphe; si G E 9 alors V@EOG,
(supposons @[x,, a,]= @[x,, a,]). Comme prtctdemment soit G’ obtenu a partir de G en supprimant [xl, a,], [x,, a,,] et en remplaGant par [a,, xp] et [a,, x,] on @[a,, x,] = @[a,, x,] et donc H E9.Pour construire un snark aura aussi V @ E aG., il suffit alors de s’assurer que H n’a ni triangle ni carrt et qu’il est bien cycliquement 4 ar&tesconnexe. Remarquons que la construction ci-dessus s’eff ectue aussi avec d’autres’snarks que le graphe de Petersen. On peut donc ainsi crier une infinitt de snarks. I1 semblerait donc prtftrable, pour rendre son mystkre a la chasse aux snarks, de les appeller des k-snarks si et seulement si ce sont des snarks cycliquement k-aretes connexes. L’intCrCt du problkme devenant alors la recherche des 5-marks, 6-snarks,. . . . Signalons enfin que, s’il est possible de construire des 4-snarks avec S(G) aussi grand que l’on veut, nous ne connaissons pas pour le moment de k-snarks ( k a 5 ) ayant S ( G ) a 3 .
Fig. 4.
28
J.-L. Fouquet
Bibliogaphie [l] Bussemaker, CobeljiC, CvetkoviC et Seidel, Cubic graphs on s14 vertices, J. Combin. Theory 23(B) 234-236. [2] J.-L. Fouquet, Jolivet et Riviere, Graphes cubiques d’indice 3, graphes cubiques isochromatiques, graphes cubiques d’indice 4, i paraitre. [3] M. Gardner, Mathematical games, Sci. Am. 234 (1976) 126-130. [4] R. Isaacs, Infinite families of non trivial trivalent graphs which are not tait colorable, Am. Math. Monthly 82 (1975) 221-239. [5] C. Payan, Thkse d’Etat, Universiti de Grenoble (IRMA) (1977).
Annals of Discrete Mathematics 9 (1980) 29-33 @ North-Holland Publishing Company
UN PROBLEME DE COLORATION AUX ASPECTS VARIES F. STERBOUL Uniuersite' de Lille 1, France Quel est le nombre minimum d'arktes d'un r-graphe d'ordre n qui, gour toute k-coloration des sommets, possede au moins une ar&e r-colorke? Cet article est c o q u comme un formulaire rkunissant tous les rksultats publiks, ainsi que quelques uns nouveaux, sans dkmonstrations, sur cette fonction S(n, r. k ) . How many edges must an r-graph of order n have, if, for every k-coloring of the vertex-set, there exists at least one r-colored edge? This paper is designed as a formulary gathering all published results, along with some new ones, without proofs, on this function S ( n , r, k).
0. Definitions r-graphe (hypergraphe uniforme de rang r (voir [l])): H = (X, $), X est l'ensemble des sommets, 8 l'ensemble des arCtes, une arCte est un sous-ensemble de cardinal r de X. k-coloration: application f surjective de X sur {1,2, . . . , k}. Pour E c X , E est dit r-color6 si If(E>l= r. X ( n , r, k): ensemble des r-graphes H dordre n (1x1= n) vtrifiant la proprittt suivante: pour toute k-coloration de X , H posskde au moins une ari5te r-colorte. S ( n , r. k ) = Min(l8l I H = ( X , %), H EX ( n , r, k ) ) (voir [lo] a [14]). Fonction de Turan [2, 4. 6, 7, 8, 16, 171: T(n, r, k): ensemble des r-graphes H d'ordre n tels que: tout A = X , avec IAl= k, contient au moins une ari5te de H.
T(n,r, k ) = Min(l811 H = ( X , a),H ET(n, r, k)). Fonction du Loto [5,15]: 9 ( n , r, k, h): ensemble des r-graphes H d'ordre n tels que: pour tout A = X, avec IAl= k, il existe au moins une arCte E de H vCrifiant \ A n El 2 h. L(n, r, k, h ) = Min((81I H = ( X , a),H E X ( n , r, k, h ) ) .
1. Formules generales S ( n , r, k )
S ( n , r, k - 1) - 1,
S(n,r, k ) s S ( n , r , k - l ) - l ( k - 2 ) / ( k - r ) ] ,
S(n,r, k ) a S ( n - l , r , k)n(n-l)/(n(n-l)-r(r-l)), 29
F. Sterboul
30
S(n, r, k ) s S(n - 1, r, k - 1),
S ( n , r, k ) c S ( n , r, k - l ) ( n ( n + l)-r(r- l))ln(n+ 11, S ( n , r, k ) s S ( n - 1 , r- I , k - l ) , S ( n , r, k ) a S ( n , r - 1 , k),
S(n,r, k ) < S ( n - l , r , k ) + S ( n - l , r - l , k - l ) , S ( n , r , k ) ~ S ( p , r , s ) + S ( n - p + l , rk,- s + 2 )
pour
pssar, n-p+lak-s+23r, S ( n , r, k)Sck.,n‘-’ Ck,r
+ Ok.r(nr-2)
Cs.rCk-s+Z,r
avec (1.91,
((cs,r)”(r-2)+ (Ck-~+2.r)”(~-~))~~‘,
S(n,r,k ) = n - k + l
si n S ( r k - k - r ) / ( r - 2 ) ,
S ( n , 2, k ) = n - k + 1.
(1.10)
(1.11) (1.12)
2. Relations avec les problkmes de Turan et du Loto S ( n, r, k ) s T(n - 1 , r - 1, k - 1),
(2.1)
S(n,r,k)aL(n,r, k - l , r - l ) = L ( n , n - r , n - k + l , n - k ) ,
(2.2)
S ( n , r . k ) s T ( n , r-1, k-l)/r,
(2.3)
S ( n , r, k) 3 T(n, n - k, n - k + l)/(
n-r n-k
).
(2.4)
3. K = R S ( n , r, r ) s T ( n , n - r , n - r + I),
(3.1)
S ( n , r, r ) a S ( n - I , r- 1, r - l)n/r.
(3.2)
S(n, r, r)==2(:)/(n-r+
(3.3)
1).
4. G-systemes de triplets et topologie S (n, 3 , 3 ) = \in (n - 2)1 si 3 s n s I 2 ,
S(n,3,3)=:n(n-2)
si
n=O
ou
2(mod3).
31
Un problZrne de coloration
G-TS: 3-graphe H = ( X , S) vkrifiant la propriCtC suivante: pour tout sommet
1
x, le graphe Hx = ( X - {x}, { E - {x} E 3 x, E E %)‘ est isomorphe au graphe donnC
G (voir [3, 131). La formule (4.2) est obtenue en construisant des (Arbre)-TS appartenant a X ( n , 3,3). En particulier, l’existence des (Chaine)-TS equivaut a un plongement triangulb du graphe complet dans une surface 2 bord. La propriCtC d’appartenir Ci X ( n , 3,3) est alors voisine du lemme topologique de Sperner [9].
5. R = 3
S ( n , 3, k ) 2 Min O r d S n -k
Max( [ f n d l ; d + S ( n - d - 1,3, k - I)),
(5.1)
S ( n , 3, k)sn2/(3k-6)+ O k ( n ) ,
(5.3)
S(2k - 2,3, k ) = k,
(5.4)
S(2k - 1,3, k ) = k + 1 ( k 3 4 ) .
(5.5)
Quelques valeurs de S ( n , 3, k ) 10
3 4 5 6 7
1 3 5 8 1 2 1 6 2 1 1 2 4 5 8 9 1 2 3 5 6 1 2 3 4 1 2 3
27 2-11 7ou8 6 4
11
12
33 40 315 a17 100~11 110~12 7 8 ou 9 5 7
6. R = K = 4
n
4 5 6 S(n,4,4) 1 4 9
H
= (Z7,{(x, x
7 8 14 24 ou 25
+ 3y, x + 4y, x + 5 y ) I x E z,, y E 2,-
(0))) E %(7,4,4). (6.2)
Soit
H = ( Z , , { ( x , x + a , x + b , x + c ) ) x ~ Z ~ , ( a , b , ~ ) ~ ~ }(6.3) ) avec % = { ( l ,2,4), (1, 3,4), (1, 3,5)}. H‘ = H plus I’ar6te (0, 1,2,3) appartient
aX
(8,4,4).
F. Sterboul
32
7. Relations avec les problkmes extremaux (sous-gaphes exclus)
Definition. Si H = ( X , 8)est un r-graphe, fi est le r-graphe (X, 9,(X)- %),‘El est le (n-r)-graphe ( X , { X - E I E E ~ } )et , pour u s r , H,, est le u-graphe ( X E E % 9” (El).
u
Definition. Le c-graphe K = (X, 9)verifie la condition YS si: quels que soient Ai = X , 1 s i s s, deux B deux disjoints, tels que (Aila 2, lA1l+ * + lAsl = c + s, il existe FE9 vCrifiant I F n Ail= lAil - 1 (1 =si =s s). Definition. Nc = ensemble des c-graphes G d’ordre 2c tels que le c-graphe vCrifie pas au moins une des conditions ‘&, (lsssc).
ne
Definition. 8 x ( n ;N c )= ensemble des c-graphes d’ordre n ne contenant aucun sous c-graphe partiel isomorphe B un ClCment de Nc. Pour n 3 2c, H
E
X ( n , n - d, n - c ) e ( r H ) c )E 8 x ( n ; N c ) ,
(7.1)
S ( n , n - d, n - 1) = [(n- l)/d],
(7.2)
H E X ( n , n - d, n - 2 ) e ((CH)Jest de maille (girth) 3 5,
(7.3)
(;)/(3+
(7.4)
O,(n) 2 S ( n , n - d, n - 2) a
((;)-
0(~3/2))/(;),
~ ( nn-2, , n-2)= (;)-f(n); n3/2(2-3/2 + o(1))s f(n) s n3l2($+ o(I)), HEX(^, n - 3 , n - 3 ) j t o u t
A,
avec IAl= 5, contient au moins 4 arCtes de “ H .
S ( n , n - 3, n - 3 ) >ilgn(n
-
(7.5)
t)(n - 2).
(7.6)
(7.7)
References C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970). V. Chvatal, Hypergraphs and Ramseyian theorems, Proc. Am. Math. SOC.27 (1971) 434-440. P. Ducrocq et F. Sterboul, On G-triple systems, h paraitre. P. Erdos et J. Spencer, Probabilistic Methods in Combinatorics (Academic Press, New York, 1974). H. Hanani, D. Ornstein et V. T. Sos, On the lottery problem, Magyar Tud. Akad. Mat. Kut. Int. 9 (1964) 155-158. G. Katona, T. Nemetz et M. Simonovits, On a graph-problem of Turan (en hongrois), Mat. Lapok 15 (1964) 228-338.
Un probltme de coloration
33
[7] M. Lorea, On Turan hypergraphs, Discrete Math. 22 (1978) 281-285. [HI J. Spencer, Turan’s theorem for k-graphs, Discrete Math. 2 (1972) 183-186. [9] E. Sperner, Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928) 265-272. [lo] F. Sterboul, Un problkme extrkmal pour les graphes et les hypergraphes, Discrete Math. 11 (1975) 71-78. [ I l l F. Sterboul, A new combinatorial parameter, in: Colloq. Math. SOC.J. Bolyai 10, Infinite and Finite Sets (North-Holland, Amsterdam) 1387-1404. [12] F. Sterboul, A problem on triples, Discrete Math. 17 (1977) 191-198. [13] F. Sterboul, A problem in constructive combinatorics and related questions, Colloq. Math. SOC.J. Bolyai 18, Combinatorics (North-Holland, Amsterdam) 1049-1064. [I41 F. Sterboul. Smallest 3-graphs having a 3-colored edge in every k-coloring, Discrete Math., B paraitre. [15] F. Sterboul, Le problkrne du loto, Cahiers du Centre d’Etudes de Recherche OpCr. (Bruxelles) 20 (1978) 443-449. 1161 P. Turan, Egy grafelmeliti szelsoCrtCk feladatrol, Mat. Fiz. Lapok 48 (1941) 436-452, voir aussi: On the theory of graphs, Colloq. Math. 3 (1954) 19-30. [17] P. Turan, Research problems, Magyar Tud. Akad. Mat. Kut. Int. 6 (1961) 417-423.
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Annals of Discrete Mathematics 9 (1980) 35 @ North-Holland Publishing Company.
MESURES D E CENTRALITE D’UN GRAPHE Gert SABIDUSSI Uniuersifk de Montrial, C.P. 6128, Succ. “A”, Montrkal, Qukbec, H3C357, Canada
Abstract I1 est bien connu que le centre d’un graphe connexe G est l’ensemble des sommets x de G dont l’eccentriciti e(x) = max{dist(x, y): y E V(G)} est minimale. D’autres fonctions ont CtC utilisCes pour mesurer, d’une manibre compatible avec l’intuition, la position plus ou moins centrale d’un sommet. Une telle fonction est la centraliti m
s(x) = 1 {dist(x, y): y E V(G)} =
irq(x), i=l
OU
q ( x ) = I{y
E
V(G): dist(x, y ) = i}l.
Nous nous proposons d’btudier des fonctions de la forme m
c(x)-
2 aini(x>,
i=l
OU
a,, a 2 , .. . est une suite de reels positifs donnee, et nous imposons des axiomes
qui lient une telle fonction ? lai notion intuitive de centralitk. L‘un de ces axiomes entraine que la suite a, doit 6tre convexe, tandis qu’un autre (prbservation du centre lors de l’adjonction d‘une nouvelle argte) est relic 9 la croissance de la suite a,. Notre expos6 portera surtout sur cette dernibre question.
35
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Annals of Discrete Mathematics 9 (1980) 3 7 4 2 @ North-Holland Publishing Company
GENERALIZED RAMSEY NUMBERS INVOLVING SUBDIVISION GRAPHS, AND RELATED PROBLEMS IN GRAPH THEORY* S.A. BURR Department of Computer Science, City College, City Uniuersity of New York, New York, NY 10031, U.S.A.
P. ERDOS Hungarian Academy of Sciences, Budapest, Hungary
Let G, and G, be (simple) graphs. The Ramsey number r(Gl, G,) is the smallest integer n such that if one colors the complete graph K, in two colors I and 11, then either color I contains G1 as a subgraph or color I1 contains G,. The systematic study of r(G,, G,) was initiated by F. Harary, although there were a few previous scattered results of GerencsCr, GyBrfBs, Lehel, Erdos, and others. For general information on the subject, see the surveys [ l , 7,8]. Any notation not defined here will follow Harary [6]. Chvhtal [3] proved that if T, is any tree on n vertices, then r(T,,Kl)=(I-l)(n-l)+l. Trivially, then, if G, is a connected graph on n points, we have r(G,, K I ) 3 (I - l)(n - 1)+ 1. It appears to be a general principle that if such a graph is sufficiently “sparse”, equality holds. With this in mind, call a connected graph G, on n points 1-good if r(Gn,K,)= (1 - l>(n- 1)+ 1. We are preparing a systematic study of I-good graphs [2]. We will not discuss the results of [2], but we will mention the following interesting unsolved problem: Let Q, be the graph determined by the edges of the m-dimensional cube, so that Q, has 2” vertices, and m2,-’ edges. Is Q, I-good if m is large enough? One type of sparse graph not dealt with in [2] is that of subdivision graphs. If G is a graph, its subdivision graph S(G) is formed by putting a vertex on every edge of G. We will show that S ( K n ) ,n ==8, is 3-good. In fact, we will treat a denser graph than this. Denote by K”(n) the subdivision graph of K,, together with all the edges of the original K,. In other words, each edge of the K, is replaced by a
* This research
was partially supported by NSF grant MPE 79-09254
37
S.A. Burr, P. Erdos
38
triangle. This graph has n + (2”) = (“g’) vertices and 3(;) edges. (For consistency, we denote S(K,,) by K’(n).)We will prove the following result.
Theorem 1.’ If n
8, then K”(n) i s 3-good, that is
r ( K ” ( n ) ,K 3 )= n2+ n - 1. The proof of this theorem is somewhat long and we defer it. It appears likely that the method can be extended to show that if 1 is fixed, K’(n) is I-good when n is large enough, but we have not carried out the details. Other possible extensions are discussed at the end of this paper. We now turn our attention in another direction. Following Erdos and Hajnal [4], denote by K,,,(n) any graph homeomorphic to K,,, that is a graph formed from K,, by putting various numbers of extra vertices on its edges. The paper [4] is reproduced in [9, pp. 167-1731. Thus K,, and K’(n) are both examples of a Kto,(n). Note that a Kto,(n) has n vertices of degree n - 1 and any number of degree 2. Let Xto,(n) be the class of all Kto,(n). In [4] Erdos and Hajnal investigate the Ramsey numbers r(Xto,,(n),Xto,(n)) and r(Xtop(m),K,). (Here we have slightly extended the definition of r : If G, or G, are classes of graphs, we are satisfied if any member of a class appears in its appropriate color.) They prove (in our notation):
r(Xto,,(n),K,) > cn;(log n)-g. Our method will give, without much difficulty,
r(Xtop(4,K 3 ) < c1 d . Before we prove this, we need another result. Denote by f(n) the largest integer for which there is a graph G on f ( n ) vertices which has no triangle, and moreover every induced subgraph of G on n vertices has at least f ( n ) edges. We prove the following result.
Theorem 2.
Proof. The proof of the lower bound is implicitly contained in [4, p. 1471, so we only have to prove the upper bound. Let G be a graph with f ( n ) vertices, all of whose n-vertex induced subgraphs have at least f ( n ) edges. Let q be the number of edges of G. Then, by a simple averaging argument, we obtain
if f ( n ) 3 2 - f n l . Since G has f ( n ) vertices, it has a vertex x of valency at least n. Since G has no triangle, all the vertices adjacent to x are mutually nonadjacent.
Generalized Ramsey numbers
39
But this contradicts (strongly) the assumption that any n vertices induce at least f ( n ) edges, so necessarily f ( n ) < 2-fn4, completing the proof. Clearly, the constant 2-i could be replaced by a smaller one. However, we will not pursue this further since we believe that f(n) = o(ng),although we don't know how to prove it. We can now prove our result on r(Xtop(n),K 3 ).
Theorem 3. For some constants c and c l r cn$(log n)-g < r(Xlop(n),K 3 )< c1n:.
Proof. We have already said that the lower bound was proved in [4].We prove the upper bound by showing that r(%op(n), K 3 ) s f ( n ) + 3 n- 5 . Consider a graph G on f ( n ) + 3 n - 5 vertices such that G has no triangle. Observe that if any vertex has degree at least n in G, we are done, since otherwise we have even a K, in G. (In fact, this also is immediate from Chvbtal's result.) From the definition of f ( n ) , we see that G has a set of vertices A = {al,. . . , a,} which induces fewer than f ( n ) edges. We will develop a Kto,(n) in G for which A is the set of vertices of degree n. These vertices already span at least )(; - f (n )+ 1 edges, so that at most f(n)- 1 must be joined by other paths. We will in fact do so with paths of length two, with the midpoints being distinct, of course. Suppose, on the contrary, that we have joined k pairs of a's, k < f ( n ) - 1, but that we cannot join ai to aj by a path of length two in G which avoids all vertices already used. We have used n + k S n +f(n) - 2, leaving a set B of at least 2n - 3 vertices. Since, by our assumption, none of these are adjacent to both ai and aj in G, either aior ai is joined in to at least n - 1 vertices in B. Since we also have that q and a, are adjacent in G, we have a point of degree at least n in But this has been shown to be impossible, which completes the proof.
e.
It would be of great interest to estimate f(n), or r(XtOp(n),K3), as accurately or f(n) = o(ng).It might not as possible. At the moment we cannot prove f(n) > be out of the question to determine the existence and value of lim log f ( n)/log n. n-m
To determine the exact value of f ( n ) or r(Xtop(n), K 3 ) is probably hopeless. Now we return to the proof of Theorem 1. It is very likely that this theorem actually holds for n 3 3. Once or twice (for instance in Fact 4) we prove a trifle more than necessary in what follows in the hope that this will help eventually to fill in the missing cases.
Proof of Theorem 1. Of course, K " ( n ) has n +(;) = N vertices, so we wish to show that r(l?"(n), K3)c 2 N - 1. (That r(K"(n),K3)3 2 N - 1 follows immediately
S.A. Burr, P. Erdos
40
from the fact that K”(n)is connected.) Let G be a graph on 2 N - 1 points and assume, contrary to the theorem, that K”(n)9 G and K , 9 G. It will be convenient to make the following definition of a partial K”(n).Let A and B be disjoint sets of vertices with \A1= n and with [BIG(:).Then a K ” ( A ,B ) is any graph consisting of a complete graph on A, together with a pair of edges connecting each point of B with a different pair of points of A. Such graphs are not unique in general, but of course if IBI = (i),a K ” ( A ,B ) is a K”(n).Furthermore, if F is a K ” ( A ,B ) , define HF to be the graph with A as its vertices, with a pair of vertices joined in HF if they are joined in F through a point of B. Moreover, call a K”(A,B ) in G maximal in a given graph if there exists no K”(A,B , ) in the graph with lBll > (Bl. We will now prove a series of facts about G, leading finally to a contradiction.
Fact 1. If F is a maximal K’(A,B ) , then
fiF contains
no triangle.
To see this, assume to the contrary that a1a2a3is a triangle in fi, and let u be any vertex not contained in F. Since no two ai can be joined through u in G, u is connected to at least two ai in G. Let u, be any other vertex not contained in F ; it, too, is connected to at least two ai in G. Hence, for some ai, the edges aiu and aiu, are both in G. Since G contains no triangle, the edge uu, must be in G. But u and u1 were arbitrary vertices not in F, so these vertices span a complete graph in G. If F had as many as N vertices, F would be a K’(n);so G contains a K N , which is again a contradiction. Fact 2.
has no vertex of degree as large as L, where L
=
[in2]+ n.
Suppose that this is false; since G has no triangle, G must have a KL. Let A be a set of n vertices from the KL. Omit for the moment the other [:n2] vertices of the KL, and let F be a maximal K”(A,B ) using the remaining part of G. By Fact 1, fl, contains no triangle, so by Turin’s theorem, fi, has no more than [in’] edges, and so HF has at least (2”) -[in’] edges. Therefore, IBI 2 (2”) -[in2]. Furthermore, there are L - n unused vertices in the KL, where we have L - n = [ i n 2 ] . Therefore, we can form a K ( A , Bl), where lBll = (;), using [in’] of these unused vertices, and (2”) -[an’] vertices from B. This contradiction establishes Fact 2 .
Fact 3. Any two points of G are joined by at least 2 N - 2L - 1 different paths of length 2 . This fact follows immediately from Fact 2.
Fact 4. Let n 2 7 and let F = K”(A,B ) be maximal. Suppose that a,, u2, a3 are distinct vertices in A, and suppose that ai and a, are connected through bijEB. Let uluz be any edge in f i F . Then G does not contain all six edges of the form Ybj,.
Generalized Ramsey numbers
41
(Note that 4 = a, is permitted.) Assume this fact is false, so that G does contain s such edges. Let C be the set of vertices not in A or B, so 1CI 3iV.Let c E C. Suppose G had two edges ca, and cai. Then G would contain the two paths aicai and ulbiiu2. In F, adjoin these two and delete the path a,biiai.This new graph is a K"(A,B U{c}), contradicting the maximality of F. Thus for any c E C, there is at most one edge from c to a,, a,, a3. Therefore, at least 2N edges join the ai to C in G, and hence some ai has degree at least f N . It is easy to see that this contradicts Fact 2 if n 2 7.
Fact 5. K , = G. This fact follows easily from the well-known result that r(K,,,,K,,) S (",'"'), already proved in effect in [5]. (The paper [5] is reproduced on pp. 5-12 of [9].) We are now ready to complete the proof of Theorem 1. By Fact 5 , G contains a K"(A,8) for some A. Let K"(A,B ) = F be maximal. By hypothesis, IBI < N ; this will lead to a contradiction. Let u,u2 be an edge of I&. By Fact 3, u1 and u2 are joined by at least 2 N - 2 L - 1 different paths of length 2, the midpoints of which all must lie in A U B, by the maximality of F. Of these midpoints n - 2 lie in A. Thus 2 N - 2 L - 1- (n - 2) of these are in B, and therefore correspond to edges in HP It is easy to check that
2iV-2L
-
1 - ( n - 2 ) >[in2] if n 2 8
Because of this, some three of these midpoints correspond to a triangle a,a2a3in HF, the midpoints being of course b,,, b23rb31. But this is just the configuration prohibited by Fact 4. This contradiction completes the proof of Theorem 1. Now we prove one final result which is very simple, but interesting. Let G be a graph with 2n - 1 vertices such that K , 9 G and K,, $ G. Then G has diameter 2 . To see this, note, as we have before, that G cannot have a vertex of degree as large as n. Hence every vertex of G has degree at least n - 1. From this it is immediate that any two vertices are either adjacent or joined by a path of length 2. We close with some remarks about improvements and generalizations of Theorem 1. We have already conjectured that Theorem 1 actually holds for n 2 3 , and we have indeed proved it for n = 3. The cases 4 s n 6 7 remain open. Although the methods of this paper would certainly help, dealing with these cases is likely to be tedious without at least one new idea. A more important direction is replacing K3 by K I . Standard estimates of r(K,,, K I ) show that K"(n) cannot be I-good if I > 3, but there is every reason to believe that for each I, K ' ( n ) is I-good when n is large enough. In fact, as we have said, it should be possible to extend the proof to this case fairly directly, but we have not carried this out.
S.A. Burr, P. Erdos
42
Another interesting generalization would be to consider the subdivision graphs, or the modification we have treated here, of arbitrary graphs, rather than just K ' ( n ) or K"(n).This may be easy, but it would not be surprising if new difficulties arose. One might also consider higher-order subdivision graphs S,(K,), S,(K,), . . ,; this is probably straightforward. It may be more difficult to deal with arbitrary, but fixed, members of Xtop(n),even with the requirement that all the paths joining the n special points have lengths at least two. (Of course, some such requirement is necessary, since K, E Xtop(n),and K, is certainly not even 3-good.) One further generalization of K ' ( n ) is of interest. Let { a l , .. . , a,} be a set of vertices, and for each triple {a,, a,, a k }of them, join each to a new vertex biik. It seems certain that if I is fixed, all large graphs of this form are I-good, and similarly for the obvious generalizations. Parts of our proof of Theorem 1 generalize easily; some may not, especially those using Turhn's theorem, since these seem to need hypergraph versions of that theorem, and such versions are not nearly as precise as for graphs.
References [ 13 S.A. Burr, Generalized Ramsey theory for graphs - A Survey, in: Graphs and Combinatorics (Springer, Berlin 1974) 52-75. [2] S.A. Burr and P. Erdos, Generalizations of a Ramsey-theoretic result of Chvatal, J. Graph Theory, to appear. [3] V. Chvatal, Tree-complete graph Ramsey numbers, J. Graph Theory 1 (1977) 93. [4] P. Erdos and A. Hajnal, On complete topological subgraphs of certain graphs, Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 7 (1964) 143-149. [5] P. Erdos and Gy. Szekeres, A cornbinatorial problem in geometry, Compositio Math. 2 (193.5) 463-470. [6] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [7] F. Harary, A survey of generalized Ramsey theory, in: Graphs and Combinatorics (Springer, Berlin, 1974) 10-17. [8] F. Harary, The foremost open problems in generalized Ramsey theory, Proc. Fifth British Combinatorial Conference, Aberdeen, 197.5 (Utilitas Math. Publ. Inc., Winnipeg, Man.) 269-282. [9] J. Spencer. (Ed.), Paul Erdos: The Art of Counting (MIT Press, Cambridge, MA, 1973).
.
Annals of Discrete Mathematics 9 (1980) 43-49 North-Holland Publishing Company
A GENERAL INTERSECTION THEOREM FOR FINITE SETS Peter FRANKL CNRS, Paris and CRMA, Uniuersitk de Mantrial Let F be a family of k-element subsets of an n-set, M a k-balanced 0-1 matrix (see definition in the paper), r a positive integer. We prove that for IFl>renk-' the family F contains r members with corresponding incidence matrix M. We give some applications.
1. Introduction Let M = (mij) be a zero-one matrix with r rows and s columns. By ai, we shall denote the number of columns in which only the ith entry is nonzero. By bi, we shall denote the number of columns in which the ith entry is zero but at least two other entries are nonzero. The matrix M will be called k-balanced if (i) each of its columns includes at least one zero and at least one nonzero entry, (ii) each of its rows includes precisely k nonzero entries, (iii) ai 5 bi for all i = 1,2, . . . , r. Next, let X be a set of size n and let F be a family of k-subsets of X. We shall say that F realizes M if we can find (not necessarily distinct) members S1, S2, . . . , S, of F and distinct elements xl, x2, . . . , x, of X such that xi E Sj if and only if rn, = 1. The aim of this paper is to prove the following result with
Theorem 1. If IF1 >c(k, r)n"'
and M is a k-balanced matrix, then Frealizes M.
Remark. Note that the bound in Theorem 1 is best possible up to the value of c(k, r): the family of ( :I:) subsets of size k containing a particular element realizes no k-balanced matrix. Our proof of Theorem 1 relies heavily on methods developed in [ 5 ] ;nevertheless, it is self-contained. 2. The proof of Theorem 1 We shall apply induction on k (the case k = 1 is trivial). First of all, we shall construct a certain family G of (k-1)-subsets of X such that J G J a c ( k- 1, r)n"-'. 43
P. Frank1
44
Set Fo = F and D, = 9.Suppose that F, and D, have been constructed. If you can find a ( k - 1)-subset T of X such that T is contained in at least kr members of F,, then set Dt+1 = 0 1 U{(T,S) I T c S E 61,
F, + 1 = F, - {S I T c S E F,}. Otherwise set rn = t and stop. Since each ( k - 1)-subset of X is contained in fewer than kr members of F, and since each member of F,,, has precisely k subsets of size k - 1, we have
Hence
and
ID,(= IFI-IF,,,I>c(k, r ) n k - ' - m k - ' / ( k - l)! = c ( k - 1, r)n"-'.
For each element y of X , consider the family G = G, of ( k - 1)-subsets of X such that T belongs to G, iff (T, T U {y}) belongs to 0,. Since C IG, I = ID,), the largest of these families satisfies
Next, we shall delete a few columns from M so as to obtain a certain ( k - 1)-balanced submatrix M'. Let t be the largest number of nonzero entries found in a column of M. Without loss of generality, we may assume that the last r + 1 - t columns C, (s - r + t s j s s) of M have the following structure: (i) the column C,-,+, has nonzero entries in the first t positions and zero entries elsewhere, (ii) each of the columns Cs-r+i ( t <j G r ) is the jth column of the r x r identity matrix. Indeed, as we are about to show, this structure can be created by a suitable permutation of the rows and columns of M. In case f = 1, we note that each row of M contains k nonzero entries and so all the columns of the r x r identity matrix can be found among the columns of M. In case t 3 2, we may assume that C,-,+, has the structure specified by (i), and so b, 3 1 whenever t <j S r. Since M is balanced, we have a, 2 1, and so the jth column of the r x r identity matrix can be found among the columns of M. It is easy to verify that the deletion of the last r + 1 - t columns of M yields a ( k - 1)-balanced matrix M'. Finally, we apply the induction hypothesis to G and M': there are (not necessarily distinct) members T,, T2,. . , , T, of G and distinct elements xi (1 <j < s - r + t ) of X such that xi E T, iff mi, = 1. Recall that T U {y}E F whenever T E G.
A general intersection theorem for finite sets
45
Hence setting Si= Ti U{y} for i = 1,2, . . . , t we obtain (not necessarily distinct) members S1,Sz, . . . , S, of F. We also set x,-,+, = y. Furthermore, recall that for each TEG there are at least kr distinct elements x of X such that TU{X}EF. Note also that kr 2 s : in M, each ,row has k nonzero entries and each column has at least one nonzero entry. Hence, once distinct elements xi (1 C j < s - r + i) have been specified for some i = t 1, t + 2 , . . . , r, we can find an element x distinct from these xi’s and such that Ti U{X}EF. We set x,-,+~ = x and Si = Ti U{x}. It is not difficult to verify that xiE Si iff mii = 1 (1S i S r, 1 S j C s).
+
3. Applications Let F be a family of k-element subsets of an n-set. If F contains n o two disjoint members, then our Theorem 1 (with M having two rows and 2k elements) implies
IF\ S 2enk-’. Actually, this bound can be strengthened: Erdos et al. [4]proved that
whenever n 2 2k. More generally, if F contains no r pairwise disjoint members, then our Theorem 1 (with A4 having r rows and rk columns) implies
IF(C renk-’. Again, this bound can be strengthened. Erdos [3] proved that
whenever n 3 no(k, r). The best value of n J k , r ) is not known. Bollob2s et al. [l] proved no(k, r ) s 2 k 3 r , the author can prove no(k, r ) S ckr2 but the question whether no(k, r ) ~ c ’ k remains r open. The lower bound no(k, r ) z kr is obvious. For n L kr, the bound
has been established by the author (unpublished). Let A be a family of subsets of [l, I ] = {1,2,. . . , r}. We shall say that a family F of k-subsets of an n-set represents A if there are (not necessarily distinct) members S , , S 2 , . . . , S, of F such that
n S i # 4 iffIEA. iel
46
P. Frank1
Clearly, if any family F represents A, then (i)
I' c Z E A implies I' E A.
Furthermore, if k s l , then (ii)
{ i }A ~
for all i = 1 , 2 , . . . , r.
We shall assume that A has the properties (i), (ii) and (iii)
(1, 2 , . . . , r } & A :
if (iii) fails while (i) holds, then any nonempty family (with k 2 1) represents A. By f(n. k, A), we shall denote the largest size of a family which does not represent A. Since the family of all the k-subsets containing a particular element does not represent any A with the property (iii), we have
f(n, k , A ) a (n - 1 k-1
)
for all such A.
Theorem 2. If k 3 (cri21), then f(n, k, A )< c(k, r)n"-'.
Proof. It will suffice to construct a k-balanced matrix M = ( r q i ) such that the sets S1,S 2 , . . . ,S, defined by
si= { j I mii = 1) satisfy (1): if I F ( a c ( k ,r ) n k - ' , then, by Theorem 1, F realizes M and so F represents A. To construct M, consider the family A* of maximal subsets of A ; by Sperner's theorem [ 7 ] ,
and so IA*I S k. Now consider the matrix M' whose columns are the incidence vectors of the members of A*, with a! and 6; as defined in the introduction and with c; standing for the number of nonzero entries in the ith row. Since
b! S I{I E A* 1 i & I}\ and c ; = ~ { Z EA* 1 i E 01,
we have b; + c; S IA*(S k for all i. Hence the desired k-balanced matrix M can be obtained by adding to M' a suitable number of columns of the r x r identity matrix: k - c; copies of the ith column for each i. Let us show at once that the assumption k s(c,;21) in Theorem 2 is best possible for r odd and nearly best possible for r even. For every choice of n, r and k such
A general intersection theorem for finite sets
that
We shall construct a family F of k-subsets of an n-set X such that
IF1 a [ n / k l k and yet F does not represent A
= {Ic [ 1, r ]
1 111 [ir] + 1).
To do so, we partition X into pairwise disjoint subsets XI, X,, . . . ,Xk of size at least [n/k] and then set S E F iff )SnXil= 1 for each i. Assuming that F represents A we find members S , , Sz, . . . , S, of F such that
In particular, for each subset I of [l, r] such that ) I )= [ i r ] + 1 there is a subscript s such that
xs n (
si)
+ 4.
Now (2) and the pigeon-hole principle imply the existence of distinct sets I, J such that (I) = ( J (= [ir]+ 1 and such that
+
for some subscript s. Since 11) ) J (> r, there is a subscript j Si n Xs by {y}, then YE
E
I n J. If
we denote
n Si
iEIUJ
contradicting (3). If c(A) denotes the smallest number of members of A whose union covers [ 1, r], then
The proof is easy: consider a set X of size n along with a subset Y of size c(A) - 1 and the family F of those k-subsets of X which meet Y.Clearly, F contains no members S1, Sz, . . . , S, satisfying (1): if it did then each of the c(A)- 1 sets Iy={iIy~Si}( Y E V whose union is A would belong to A. In fact, we conjecture that
P. Frank1
48
whenever k 2 k , ( A ) . We can prove this conjecture in case
A = { I c [ 1, r ] 1 )I1S t } for all r and t but the proof is rather involved. Many extremal problems can be expressed in terms of f(n, k, A ) . For instance, consider the problem of determining ex(n, G), the largest number of edges in a graph with n vertices and without a forbidden subgraph G. With the edges of G enumerated as e l , e 2 , . . . , e, we set
Since AG determine G, we have ex(n, G) = f(n, 2 , A G ) . Unfortunately, our Theorem 2 does not apply in this context: the inequality k (&,) rules out the nontrivial cases. A number of interesting problems arises in connection with
A(r, t ) = ( I C [ l , r ] l )I)=s~}. For instance, the theorems of Erdos-Ko-Rado and Erdos quoted above may be stated as n-1 f(n, k, A ( 2 , 1))= ( k - 1) whenever n 2 2 k and whenever n 2 n,(r, k ) , respectively. ChvCital [ 2 ] proved that f(n, k, A ( k , k - 1))=
(nk -- 11 )
whenever n > k + 1.
Note that our Theorem 1 with M = [I,J - I ] such that I is the k x k identity matrix and J is the k x k matrix filled with ones implies
f(n, k , A ( k , k - l ) ) < e k n k - ' . ChvCital and Erdos [ 2 ] conjectured that f(n, k, A(r, r - 1))=
(n - 1 -
r- 1 whenever r < k sn. r
This author [6] proved that
our Theorem 2 implies only f(n, k, A(r,r - 1))< ernk-'
whenever k
([r/21).
A general intersection theorem for finite sets
49
Finally, we can prove that
~ ~ n ~ ' ~ =3, ~A(4, f ( n2 ), ) S ~ ~ n " ' ~ and conjecture that
is the correct exponent.
References [l] B. Bollobh, D.E. Daykin and P. Erdos, On the number of independent edges in a hypergraph,
Quart. J. Math. Oxford (2) 27 (1976) 25-32. [2] V. Chvatal, An intersection theorem for finite sets, J. London Math. Soc. 9 (1974) 355-359. [3] P. Erdos, A problem of independent r-tuples, Anales Univ. Budapest 8 (1965) 93-95. [4] P. Erdos, Chao KO and R. Rado. An intersection theorem for finite sets, Quart. J. Math. Oxford (2) 12 (1961) 313-320. [5] P. Frankl, A constructive lower bound for Ramsey numbers, Ars Combinatoria 3 (1977) 297-302. [6] P. Frankl, On a conjecture of Chvital and Erdos, to appear in J. Combin. Theory (A). [7] E. Sperner, Ein Satz uber Untermengen einer endlichen Menge, Math. Z. 27 (1928) 544-548.
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Annals of Discrete Mathematics 9 (1980) 51 @ North-Holland Publishing Company.
A NEW PARADIGM FOR DUALITY QUESTIONS Michael 0. ALBERTSON Smith College, Northampton, ME 01063, USA
Abstract Given an incidence system S, form a graph G = G(S) by joining two objects +A, with an edge if they occur in a common block. For k = 1,. . . set A1 + (A{+ * * +AL) equal to the maximum number of vertices in a k-colorable induced subgraph of G (G'). The sequence A l , . . . , Ax (A{, . . . , A;) are called the partition sequence of S where x (0) is the chromatic number of G(G"). Many duality theorems (e.g. Dilworth's Theorem and Konig-Egervary's Theorem) can be expressed as x = A ; or @ = A l . Greene and Kleitman showed that for the incidence system generated by a poset (where blocks correspond to chains) the partition sequences are monotonic and the duality theorems are a special case of the conjugacy of the partition sequences. Albertson, Berman and Shearer have established necessary conditions for a sequence to be a partition sequence of some incidence system. These results suggest a number of interesting questions for various incidence systems. Partial answers for those questions associated with the Konig-Egervary theorem will be discussed.
51
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Annals of Discrete Mathematics 9 (1980) 53-62 @ North-Holland Publishing Company
THE COMPLEXITY OF A PLANAR HYPERMAP AND THAT OF ITS DUAL Robert CORI and Jean Guy PENAUD U.E.R. de Mathimarique et Informatique, Laboratoire Associi au C.N.R.S.. Uniuersiti de Bordaux I , 3.51, Cours de la Libiration. 3340.5 Talence, France The number of spanning trees in a graph is often called it's complexity [I]. A tree is of course of complexity one and it is a classical result of Cayley that the complete graph K , has complexity n"-*. Between these two numbers lies the complexity of a connected graph. In the case of planar maps it is well-known that a map and its dual have equal complexity (see for instance [ 2 ] ) . In this communication we show that a similar result holds for hypermaps. To prove this result we use a diagram containing six hypermaps which is very related t o W.T. Tutte's Trinity [9]. All the needed definitions are recalled. We thus give a few details about hypermaps. their underlying hypergraphs and the maps associated t o them. The definition of a spanning hypertree is also given.
1. What is a hypermap Let [ n ] denote the subset {1,2,. . . , n } of positive integers.
Definition (see [3]).A hypermap is an ordered pair of permutations (a,a),which generate a transitive group on [ n ] .To verify this condition one can construct a graph with vertices {1,2,. . . , n } and with edges (a, a a ) and (a, a a ) for all a in [ n ] .The graph is connected if (a,a) is a hypermap. The underlying hypergraph of (a,a) is obtained as follows: take as set of vertices the orbits of a ; each orbit of a determines then an edge consisting of all the vertices intersecting it.
Example. Let 2 = (a,a) be defined as follows: a = ( 1 , 2 ) ( 3 , 4 , 5 ) ( 6 , 7 ,8)(9, 10)(11, 12, 13), a = (1,5)(2, 11,8,4)(3,6,9)(7,12, 10)(13).
Then its underlying hypergraph has five vertices (a, b, c, d, e ) and five edges {a, b}, {a, b, c, e ) , {b, c, 4, {c, d. el, {el.
Of course a given hypergraph may be represented in several ways by a hypermap. If a is a fixed point free involution, then (a,a)is said to be a map and this kind of combinatorial maps is now well-known [ S , 61. 53
R. Cori, J.G. Penaud
54
In order to represent topologically a hypermap, first draw disjoint circles on the plane as much as u has orbits; and put on each circle the different elements of an orbit of u in a conventional order (say clockwise). This is what combinatorialists often do to represent a permutation. To this set of circles add closed curves to represent the orbits of a : each curve contains all the elements of an orbit of a (and these elements are met counter-clockwise). Fig. 1 is a topological representation of the hypermap (u,a ) given above. The orbits of u are called the uertices, those of a the edges and of a - l u the faces of the hypermap.
Fig. 1.
A natural question is to know if it is possible to draw the figure such that the domains delimited by the circles and the close curves intersect only on the points corresponding to 1 , 2 , . . . , n. Using the topological theory of maps and hypermaps it is not too difficult to prove that a necessary and sufficient condition for it is that the characteristic x(u, a ) of the hypermap is equal to 2. The characteristic x(u,a ) is given by
+
x(u,a ) = o(u) + o ( a ) o ( u - ' a ) - n
where o ( u ) denotes the number of orbits of u. When x(u,a ) = 2, the hypermap is said to be planar. Two hypermaps (a,a ) and ( u ' , a f )are isomorphic if there is a one-to-one mapping cp from En] onto [ n ] such that ITf = c p u c p - 1 ,
a ' = cpacp-1.
They are said anti-isomorphic if (u, a ) and (up', a ) are isomorphic. 2. The diagram of the six hypermaps
To form the dual of a map one exchanges the faces and the vertices, the edges remaining the same for both of them. We have a similar notion for hypermaps.
The complexity of a planar hypermap
55
The dual of X = (a, a) is the hypermap X* = (a-lcr, a-').Of course the dual of X" is X,as (a-')-'a-'a= aa-la = a ; also it is easy to verify that X* is planar when X is (use the fact that a and a-' have the same orbits). The reciprocal of a hypermap X = (a, a) is = (a.a),this notion is similar to that of "line graph" for a graph. The reciprocal of a planar hypermap is also planar as 6 l a ( = ( a - * a ) - ' ) and a p l uhave the same orbits. Taking the dual and the reciprocal of a hypermap and so on, one can find six hypermaps given in the following diagram: (a,a )
reciprocal / (a,a)
(a- 1 a,
- 1)
dual
reciprocal
(a-'a, a - l ) re c i p r o c a h
d u a y
(+-'a)
(a-I,
a-'a)
(a-l,
It is easy to verify that all the six hypermaps of the diagram are planar when (a, a) is. We call (a-', a - l a ) the hyperdual of (a, a);and we verify that the hyperdual of (a-', (+-'a)is (a, a). Using this diagram, we will show that to any hypermap X we can associate a map % with the same characteristic, in which the degrees of vertices are even and the faces have all degree three. Furthermore, if X is planar % is an even triangulation. To prove this we need to explain a few notations and to define how to associate a "bipartite map" and a "reduced map" for a given hypermap.
Notations. Let B a finite set and T a permutation acting on B ; we call B a disjoint copy of B, p the natural bijection from B onto B ( p ( b )= b; p ( b ) = b ) ; ii the permutation acting on B verifying f = p r p and for two permutations T, and T , acting on disjoint object sets B , and B2 we use (see [lO,p. 161) the sum T ~ @ T which , is the permutation acting on B1U B2 as follows: r , ( b ) if b E B,, ~ , ( b ) if b E B,
and we note (a,a,B ) the hypermap (a, a) acting on the object set B.
Bipartite map. For every hypermap X = (a, a,B), Walsh [lo] has defined the bipartite map Bip(X) which has the same characteristic, by the ordered pair of permutations a @ E - ' , p, acting on B U B. If X has a planar representation we can obtain that of Bip(X) by putting a new vertex in each edge and each vertex of X and joining each two new vertices if they are respectively in a vertex and an edge of X which intersect. As uE16-l and ( a @ F ' ) - ' are conjugate, it is clear that Bip(X) and Bip(X') are anti-isomorphic when X and 2' are two reciprocal hypermaps.
R. Cori, J.G. Penaud
56
Reduced map. In an another way, as described in [7], we can define the reduced map Red(%), which has the same characteristic as X , by the permutation 8 and the involution /3 acting on B U B as follows: __
~
O(b)= d b ) ,
P ( b )= a ( b ) ,
O(b)= b,
P(b)= a - ' ( b ) .
In this case we obtain the planar representation of Red(%) from that of X by putting a new vertex in each vertices of X and joining them by an edge iff the vertices which contain them are encountered consecutively on the same edge of X. Example. The hypermap X B
= (u,a,B )
of Fig. 1 where
= [13],
u = (1,2)(3,4,5)(6,7,8)(9,10)(11, 12, 13), (Y
= ( l ,5)(2, 11, 8,4)(3,6,9)(7,12, 10)(13).
Then Bip(X) is the map of which the permutations acting on B U
are
T = ( i , 2 ) ( 3 , 4 ,5)(6,7, 8)(9, i o ) ( i i , 12, i3)(i, 5 ) x(2,;i, =(I,
Ti, ii)(S, c), 6)(5,lo, 12)(13),
i)(2,2)(3,3)(4,4)(5,5)(6,6)(7,5)(8,8)(9,4)(10, G)(i1,111
x (12, i?i)(13,fi).
and Red(X) is the map given by 8 and p acting on B U B as follows:
s, 8)(4,9,fi,10)
8 = (7, 1,2,2)(3,3,?,4,5, 5)(6,6,7,7,
x (ii, ll,E,12,iii, 13),
p = (i,5)(5, i)(2, i i ) ( i1,8)(8,4)(4,2)(3,6)(6,4)(9,3)(7,12) x ( 12, i6)(10, ?)( 13 , E ) .
Fig. 2 and Fig. 3 give the representation of Bip(X) and Red(%). Note that to obtain the representation of Red(%) from that of % one has simply to reduce to a single point the circles which represent the orbits of (T. These two following remarks will explain the relationship between the bipartite map and the reduced map of a hypermap and they will be useful to prove the property above announced:
Remark 1. The reduced map of a hypermap X k = (u-',u - ' a , B) are two anti-isomorphic maps.
= (u,a,B )
and its hyperdual
Remark 2. The bipartite map of X and the reduced map of the hypermap X* = (a-'u, u p ' ,B) are two dual maps.
57
The complexity of a planar hypermap
Fig. 2 .
With the diagram of the six hypermaps we can then associate the three bipartite maps and (due to Remark l), the three reduced maps which are (as written in Remark 2) the dual maps of the three first.
Proof of these remarks. To simplify the writing of this proof let us introduce the following notation: for any permutation .rr acting on B, 6 and ?r are the permutations acting on B U B as 'follows: if b E €3, if
6~ B,
G ( b )= 6; + ( b ) = p.rr(b)= .rr(b), G(b)= .rr(b); +(b)= b.
Briefly to obtain G (respectively +) simply write 6 after (resp. before) each b in every orbit of rr. Then Red(a, a ) = (6, p, B U B) where 13= 6 ;p = TI,, (b, (yo) (where KI denotes the product of the orbits). Similarly Red(a-', C l a )= (Of, p', B U B) verifies 8' = a-1 p'= L
nbeB
( b ,a - ' a ( b ) ) .
Fig. 3.
R . Cori, J.G. Penaud
58
Let p be the one-to-one mapping from B U B defined by p ( b )= b, p ( b ) = a(b); it is easy to verify that p is an isomorphism of maps between (8, p, B U B) and (&I, p', B U B) and the first remark is proved. By the same way we prove the second remark: let ( ~ , 6 and ) (8, p ) be the pairs of permutation of Bip(u, a ) and Red(a-'u, u-I); so we have L
7
= V@cY-';
6=p;
8 = a-'U;
p = R ( b ,a - ' ( b ) ) t
and the permutation p8 of the faces of Red(a-'u, a - l ) is equal to a@? then to T. If we introduce the bijection v from B U B as follows: v(b) = b, v ( b ) we shall prove the second remark by verifying that v is an isomorphism of map between ( ~ 6 and ) (PO, p).
=a),
Derived map of a hypermap. This notion generalizes the "derived map" of a map introduced by Tutte in the paper "Duality and trinity". Given a hypermap X = (u,a,B ) we will construct a map qD= (T, 0, B') in the following way: - B'= B l U J UB4 U BsU B, where the Bi's are six disjoint copies of B - 7 = a@a-'@Ua-' where 6 is obtained from u by replacing in each orbit b by bl, b,, a - l from a-' replacing in each orbit b by bZ,b4 and aa-l from cap' replacing 6 by b3, b,
Gz
d
L
-p = 11btB (bl,
b4)(b2, b S ) ( b 3 ,
It is easy to verify that
and then x(a, a ) = X ( T , (3). The orbits of T having an even number of element and these of PT having exactly three elements, the topological representation of Y D is an even triangulation. To picture this triangulation first put a point in every vertex, every edge and every face of the representation of 2 and join two points if the elements of X containing them intersect. Now we can easily verify, but write the proof is fairly fastidious, the following theorem:
Theorem. Every even triangulation is the derived map of a planar hypermap, the three reduced maps of which form the "Trine alternating maps" of Tutte. 3. Spanning hypertrees and codes A hypermap ( a , a ) is a hypertree if it is planar and has only one face (o(a-'a)= 1). In this case the underlying hypergraph is connected and has no cycle [3]. And this is a necessary and sufficient condition.
59
The complexity of a planar hypermap
Given a hypergraph H it is not always possible to delete edges to obtain an hypergraph H' verifying: H' has no cycle and is connected. (See for instance the underlying hypergraph of Section 1 .) To define a spanning hypertree we introduce the refinement of a hypergraph in the following way: H ' = ( X , 6') is a refinement of H = ( X , 6) if each edge of 6 is a disjoint union of edges in 6' and each edge of 6' is included in an edge of 6. Let (a,a ) be a hypermap; (a,a ' ) is a refinement of (a,a ) if a = ala2* . * ap, a ' = a ; a ; * * * a ; when the ai are cyclic, each a! acts on the same set as ai and x(ai,a!)= 2. As o ( a i )= 1, one has also, if n denotes the number of elements of the object set of a and a, 2p = o ( a )+ o(a')+o(a'-'a) - n hence o (a ' )+o(a'-'a)= p + n. A spanning hypertree of (a,a ) is a refinement (a,a ' ) which is a hypertree. The a). number of spanning hypertrees of (a,a ) is the complexity of (u, A code of a hypermap (a,a ) is a cyclic permutation { such that x({, u)= x ( t a )= 2. It is shown in [3] that a necessary and sufficient condition for a hypermap to be planar is to possess a code. These codes are also used there to represent a hypermap by a word. It is announced in [4]the following theorem which we will prove in the next section:
Theorem 1. The number of codes of a hypermap is equal to the complexity of its hyperdual. Let us now use this theorem to obtain the result above announced.
Theorem 2. The complexity of (u,a ) is equal to that of
(a-la,
a-').
Proof. As the hyperdual of ( a - ' ,a - ' a ) is (0;a ) the complexity of (a,a ) is (by Theorem 1) equal to the number of codes of (a-',u-'a).By the very definition of a code each code of a hypermap is also a code of its reciprocal. Then the complexity of (a,a ) is equal to the number of codes (a-'a, 6'). But applying Theorem 1 once more, it is also the complexity of its hyperdual (a-la, a - ' ) and the Theorem 2 is proved.
4. Codes and complexity The construction of all the codes of a planar hypermap is given by the following lemma. Let us first give a few definitions: A transposition on [n] is a permutation which fixes all but two points i and j of [n]; we will denote T ~ such , ~ a transposition. The number of its orbits is clearly n-1.
R. Cori. J.G. Penaud
60
A transposition T ~ is, said ~ to intersect a permutation u if i and j are in different orbits of u.In that case UT,,~has an orbit less than o. A transposition T ~which , ~ do not intersect a permutation u is said to be included in cr; in that case the number of orbits of U T ~ ,is~ one more than that of u.
Lemma. 5 is a code of the planar hypermap (u,a ) i f and only if there exist o(a)- 1 = p transpositions r , , T ~. ., . , rp such that T~ intersects UT,,T~. * . T ~ and - ~ is included in ( Y - ’ U T ~ T *~* . T ~ for - ~i = 1 , 2 , . . . , p and 1;= U T ~ T . .~* T ~ . The proof of this lemma requires the following fact: given a permutation a,the set of all circular permutations 1; satisfying ~ ( 5a) , = 2 verifies: there exist o(a)- 1 = p transpositions T ~T , ~. ., . , T~ such that for any i, T~ intersects U T ~ T * ~* T , - ~ and 1; is equal to U T ~ T . .~* T ~ This . fact can be proved easily (see for instance [3] and use the notion of “mot emboitk”). To prove the “if” part of this lemma, remark that the second condition implies that o(a-’1;) is equal to O ( ( Y - ~ C T ) + also ~ equal to ~ ( a - ~ u ) + o ( u1,) -then ~ ( 5a) , = x(u,a)= 2. It remains to prove the “only if” part; let 1; be a code of (a.a). As x(& o)= 2 we have = m1* * * T~ with the T ~ ’ Sverifying the first condition. We have +
<
~ ( u T , ,a)= O ( U T , )
X(UT,,
+ o ( a )+ o(a-lu~J- n,
a)= x(o, a)+ o ( m l ) io ( a - l m l )- o(a-’v)- o(u),
X(6, a)=X(ml *
‘ ’
. ‘ . TP-1)
TP-i, ~ ) + O ( ~ ) - O ( @ T I
+ o(a-’<)- o ( a - l m l . . r p - l ) . Then X ( 5 , a)= X ( U , a)+ 1- o ( U ) +
P
1
(O(U-’UTl
’
.
*
T p )- O ( ( Y - l U T 1
.
* ‘
Tp-l)).
i=l
As (1;. a) and (u,a) are planar and p = o(u) - 1, we deduce that the sum is equal to p; hence each term is equal to p and the result is proved. From this lemma one can deduce the following:
Theorem. There is a bijection between the set of codes o f a planar hypermap and the set of spanning hypertrees o f its hyperdual. Let (u-’, 0 ) be a spanning hypertree of (up‘, @-la):we are going to show that crB is a code of (u, a).As (K’, 0) is a hypertree, K’a-’ has exactly one orbit and its inverse a0 too. Further more ~ ( u -0’) = , 2 and it is also true that
x ( d , U ) = o ( d ) + O ( U ) + o(e)-
n = x(u-’.
e) = 2.
The complexity of a planar hypermap
61
Let us prove now that x(u8, a ) is also equal to 2. Computing x(u8, a ) one finds
x(ae,a ) = 1+ o ( ( Y ) +o ( a - b e ) - n. As 8 is a refinement of u-'a we also have: n + o((+-la)= o(0) + O ( e - ' u % ) and
x(ue,
a ) = 1 + ~ ( a+ o) ( u - ' ~ ) o(e).
From the planarity of (u, a),
+
o ( u ) + o ( u - ' a ) o ( a )= n + 2
and from the planarity of (u8, u) o(U)
+ o(ue)+ o(e) = n + 2.
Thus we obtain
x(ue, a ) = 1+ o ( ~ o=)2 as a8 was found to have one orbit. We have thus proved that u0 is a code of (a,a ) . Conversely let 1; be a code of (u, a ) ;the lemma insures the existence of the transpositions T ~ T, ~ . ,. . , T ~ Let . us denote by 0, the product T1T2 . . T ~ then ; O0 is the identity, and let 8 be an abbreviation for $. We will prove that 8 is a spanning hypertree of (up', a-'a). First it is easy to verify that 8-'u-' = (u8)-' has only one orbit, and, of course, x(B-'o-', 6') = x(a,8, a ) is equal to 2 as a8 is a code of (u, a). Let us show now by induction on i that 8, has exactly n - i orbits; clearly for n = o it is the case. As u8 is a code, x(u,8) = 2 and o ( e ) + o ( u ) + 1 - n = 2. Moreover p = o(u)- 1 as each
T,
intersects d - , and u8 has one orbit; thus
O ( 8 ) = n - p.
Let us denote by
E,
the difference o(8,)-o(8,-,);
E,
is equal to + l ; and
i=l
then si is necessarily equal to -1 for any i. It is now possible to end the proof in verifying that 8, is a refinement of (+-'a by induction on i. The case i = 1 is clear as is included in u-la. Let p l , p2, . . . , p, be the different orbits of (+-'aand suppose that OiPl = p:p; * * pi such that the pJ acts on the same object sets as the pi and that x(pi, p;) = 2. As T~ is included in a-'aOi, it is a fortiori included in a-lu and in its inverse u-la. Let p1 be the orbit of u-'a which contains 7,.Then we have 0, = p'lp; . . * Pi
R. Cori, J.G. Penaud
62
where py = p ; ~ ~ . Let us compute x ( p l , py):
Remark. We can verify that, in the case of planar maps, the two trees, which span two dual maps and correspond to the same code 6 are two complementary trees. Indeed let X = (a,a ) and X* = (a-la, a - ' ) be two dual planar hypermaps and let 9 be a spanning hypertree of X.Then .T = (a,8) where 8 is a refinement of a. Thus 5 = a - ' 8 is a code of (a-',@ - ' a )and also of the reciprocal (a-'a, a-I). Hence 9'= (a-'a, 8') is a hypertree which spans X* with (a-la)-lc
= a-1aa-'8
= a-18.
In the case of planar maps, a is a fixed point free involution. If (b, 6) is an edge of 3, then (b, F) is an orbit of 8 and of a. Thus 8'(b)= a-'O(b) = a-la(b) = b and ( b . 6) is not an edge of 9: 5 T is the complement of 3 (see [2]).
References [I] [2] [3] [4]
N. Biggs, Algebraic Graph Theory (Cambridge University Press, London, 1974). N. Biggs, Spanning trees of dual graphs, J. Combin. Theory 11 (B) (1971) 127-131. R. Cori, Un code pour les graphes planaires et ses applications, Astirisque 27 (1975). R. Cori, Codage d'une carte planaire et hyperarbres recouvrants, Colloques Internationaux du CNRS, No. 260, Problkmes Combinatoires et Thkorie des graphes, Orsay (1976). [5] J.R.Edmonds, A combinatorial representation for oriented polyhedral surfaces. Notices Am. Math. Sci. 7 (1960) 641. [6] A. Jacques, Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Paris Ser. A 267 (196X) 625-627. [7] J.G. Penaud, Quelques propriitks des hypergraphes planaires, Thesis, Bordeaux (1974). [8] J.G. Penaud, De la planariti de certains hypergraphes, C. R. Acad. Sci. Paris Ser. A 277 1973) 93 1-933. 191 W.T. Tutte, Duality and trinity, infinite and finite sets, Colloq. Math. SOC.Jhnos Bolyai 10 1973) 1459-1472. [lo] T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory 18 (B) (1975) 155- 63. [ I 11 A.T. White, Graphes, Groups and Surfaces (North-Holland, Amsterdam, 1973).
Annals of Discrete Mathematics 9 (1980) 63-64 @ North-Holland Publishing Company
ACCEPTABLE ORIENTATIONS OF GRAPHS Christopher LANDAUER" Language Research Laboratory, Paffern Analysis and Recognition Corporation, 228 Liberty Plaza, Rome, N Y 13440, U.S.A. This paper solves a problem of E. Howorka on undirected graphs: every finite graph has an orientation for which every edge e = ( a + b ) satisfies o ( a ) - o ( b ) c 1 (where o(u) is the outdegree of the vertex u ) . The result was proved after computer experiments with a primitive heuristic showed that such orientations were relatively easy to find.
At a recent Combinatorics meeting, Professor Edward Howorka asked whether or not every finite, undirected graph has an acceptable orientation (see definitions below). It turns out that not only do such orientations always exist, they are easy to find.
Definition. An edge e = ( a + b ) of a directed graph is acceptable iff o ( a ) - o ( b )G 1, where o ( a ) denotes the outdegree of the vertex a . An orientation of an undirected graph is acceptable iff every edge is acceptable (we identify an orientation of an undirected graph with the corresponding directed graph). An edge e = ( a + b ) is critical iff o ( a )- o ( b )= 1 . These edges are the objects of study during the proof of the answer to Howorka's problem.
Theorem. Every undirected graph has an acceptable orientation. Proof. By induction on the number of edges. Graphs with few edges are easy to check. Graphs whose edges are all loops are easy to check. Let { a , b } be any nonloop edge of the graph (i.e. a # b ) . Find an acceptable orientation of the graph obtained by deleting this edge. Suppose o ( a ) ~ o ( b ) . Let a = c,, -+ c , + . * . + c, be a maximal sequence of critical edges from a, so r 3 0. Add the edge a + b, and reverse every edge ci+l+ ci, 1 s i s r. Then the outdegree of every vertex is the same (except c r ) . The new edge a + b is acceptable, since o ( a ) < o ( b ) (unless r = 0, whence o ( a )< o ( b )+ 1 already). The edges ci+c,-, are acceptable, since
o ( c i ) - O ( c i - , )=
[
-1
for i < r ,
0 for i = r. All edges c, + d are acceptable, since none were critical before.
* Present address: System Development Corporation, 2500 Colorado Ave., Santa Monica, CA 90406, U.S.A. 63
64
C. Landauer
We have therefore produced an acceptable orientation for the graph. Moreover, as a result of some computer experiments on graphs with over 100 vertices, we have an (easy) algorithm for producing an acceptable orientation “quickly”, relative to the number of edges of the graph. In order to describe the algorithm, we need some definitions.
Definition. For an edge e = ( a + b ) , write A(e) = o ( a ) - o ( b ) and where i(u) is the indegree of the vertex u.
r(e)=i(b)-i(a),
Lemma 0. If edge e is reversed to produce edge e ’ = ( b + a ) , we find A’(e’)= 2 - A ( e ) , T’(e’)= 2 - T ( e ) , where the primes denote the new graph (with e reversed). Lemma 1. The total change in S , = c (eEE)(3A(e)-I‘(e)) obtained by reversing edge e is T,(e) = 8 -8A(e). The algorithm that find an acceptable orientation begins by minimizing S, in O(lE1‘) time, as follows: find the edge e with smallest T,(e), and reverse it; repeat until every T , ( e ) s O . Then we are done. The author is indebted to the referee for the form of Lemma 1, which allows an easier formulation of the algorithm. The actual number of steps required is much less than lEI2. For graphs with 4000, 1000, 400 edges and 100, 40, 40 vertices (respectively), the maximum number of inversions required was less than 180, 60, 50 (respectively), and the average numbers were less than 150, 40, 35. These averages are for hundreds of randomly generated graphs, but with a slightly different algorithm than the one described above.
Annals of Discrete Mathematics 9 (1980) 65 @ North-Holland Publishing Company.
AN ALGEBRAIC UPPER BOUND ON THE INDEPENDENCE NUMBER OF A GRAPH L. LOVASZ Bolyai Institute, Aradi vkrtanrik tere 1 , H-6720 Szeged, Hungary
Abstract This upper bound on the independence number of a graph has been introduced to study the so-called Shannon capacity. It is defined by linear algebra and eigenvalue techniques. In this talk we discuss some other aspects including a new characterization of perfect graphs.
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Annals of Discrete Mathematics 9 (1980) 67-78 @ North-Holland Publishing Company
PREUVE ALGEBRIQUE DU CRITERE D E PLANARITE D E wu-LIU P. ROSENSTIEHL Directeur d’Etudes, EHESS, 54bd Raspail, 75006 Paris, France W. Wu and Y. Liu have sketched topological proof of a planarity criterium, which consists of associating to each crossing of an embedding of a graph G in the sphere an edge of a graph H associated to G. The graph H is defined by a Tremaux tree of G (maximal depth-first-search subtree of G). The method is to test whether these edges form a cocycle of H . We give here a complete algebraic proof of their criterium, inspired by Tutte’s theory of crossings [9]. The present work can be construed as a more efficient version of Tutte’s method.
1. Introduction On entend ici par graphe G = ( V ,E ) un ensemble fini V de sommets, un ensemble fini E d’arzres, et pour toute arCte deux sommets incidents distincts, deux arCtes distinctes ayant au plus un sommet incident commun. Deux arctes ayant un sommet incident commun sont adjacenres, deux ar&tes sans sommet incident commun sont disjointes. On renvoie B C . Berge [ l ] pour les dtfinitions classiques de connexitt et 2-connexit6, et les dtfinitions des ensembles d’arCtes suivants: arbre T de G (c’est-i-dire arbre maximal), cycle, cocycle ou cobord 6s avec S c V, cycle fondamental T ( a ) associC B a $ T dont toutes les ar&tessauf a appartiennent B T. Par commoditt tout ensemble d’arCtes dCsigne aussi un vecteur dans le vectoriel sur GF(2) dont la base canonique est constituhe par E. Un arbre T de G et un sommet r E V d’un graphe connexe dkfinissent canoniquement un ordre partiel ( V , <) dont r est ClCment minimum. On appelle arbre de Tre‘maux [ 5 ]- en anglais “depth-first-search-tree’’ [8] - d’un graphe G connexe, un couple (T, r ) ou r E V et T est un arbre de G, tels que toute arCte a $ T soit incidente h deux sommets comparables selon I’ordre partiel ( V , <) induit par (T, r ) . Les arCtes qui n’appartiennent pas B I’arbre de Trtmaux sont appeltes palrnes et nottes de prtftrence par des lettres grecques. Orienter une arCte e E E de x vers y, c’est distinguer les deux sommets x et y auxquels elle est incidente, en posant x = u - ( e ) et
y
=
u+(e).
Toute arCte e E T est orientte de sorte que u - ( e ) < v’(e). Toute ar&te a $ T est orientte de sorte que u + ( e )< v - ( e ) . On pose en outre pour e E T :
V + ( e )= { x E V I u + ( e )G x } . 67
P. Rosenstiehl
68
(a) notations
(b) plongement de Tr6maux
(c) plongement non de T r h a u x
Fig. 1 . Arbres de T r h a u x de G et plongements (les arktes e E T sont rectilignes, les arktes a $ T sont curbes).
Comme T ( a )a, pour tout a$!T, au moins deux arCtes, on peut associer e ( a )E T et l’ar&te E(a)E T, telles que u - ( e ( a ) )= u+(a) et
a I’ar&te
u+(e(a)) = v-(a).
(Voir Fig. l(a).) C‘est une propriCtC ClCmentaire de labyrinthologie [7], qu’un graphe connexe admet un arbre de TrCmaux, et que l’on a les deux propriCtCs suivantes: P1 Dans un graphe 2-connexe tous les arbres de TrCmaux ont une racine de degrC un. P2 Dans un graphe 2-connexe, un arbre de TrCmaux (T, r ) &ant donnC, il existe pour tout ar&te e E T, avec u - ( e ) # r, une palme a telle que
u+(a)< v - ( e )< u - ( a ) .
Preuue alge‘brique du criDre de planarite‘ de Wu-Liu
69
Dans la suite le graphe G est toujours considCrC 2-connexe. On considere un plongement Go de G dans la sphkre [6,2], autorisant les croisements des arcs de Jordan images des aretes: points de la sphkre communs B deux aretes du plongement, sans Ctre sommets du plongement. G est planaire s’il admet un plongement sans croisements dans la sphkre. La propriCtC suivante sur les croisements permettra de ne plus s’occuper dans la suite des croisements d’arCtes adjacentes. P3 Si G admet un plongement dont les seuls croisements sont entre des ar&tes adjacentes, alors G est planaire. En effet, si deux arCtes adjacentes en u se croisent en c, en Cchangeant des deux ar6tes leurs arcs de Jordan entre u et c, on supprime exactement un croisement du plongement. On les supprime tous par itCration. On appelle plongement simple de G relativement 2 un arbre de TrCmaux (T , r ) , un plongement sans croisements sur les ar6tes de T et avec au plus un croisement entre deux palmes disjointes. On a la propriCtC suivante: P4 Pour un arbre de TrCmaux ( T, r ) de G donnC, et un plongement sans croisements To de T donnC, il existe un plongement simple de G relativement 2 (T,r ) qui prolonge P . La propriCtC P4 est essentiellement une consCquence de I’absence de cycles dans un arbre. La dCmonstration constructive consiste 2 se donner le plongement de T et de n - 1 palmes satisfaisant B la condition CnoncCe dans P4. On en considkre le graphe planaire et le graphe dual: les arCtes duales d’arcs de Jordan appartenant 2 des palmes constituent un graphe connexe, dans lequel une plus courte chaine entre deux points permet de plonger une nikme palme arbitrairement donnCe, en satisfaisant 2 la condition CnoncCe dans P4.
2. Plongements de Tremaux
Soient un plongement Go de G et r E V. Tout cycle ClCmentaire C, c’est&-dire n’incluant pas d’autre cycle, dCfinit une tripartition de la sphhre en C et deux rtgions. Si r n’est pas sur C, la rtgion qui inclut r est dite exterieure, I’autre est dite intCrieure. On note alors Exto(C) l’ensemble des sommets de Go appartenant 2 la rCgion extkrieure de C, et Into(C) I’ensemble des sommets de Go appartenant 2 la rCgion intCrieure de C. On rappelle que T ( a ) est ClCmentaire. On appelle plongement de Trimaux du graphe G relativement 2 un arbre de
P. Rosenstiehl
I0
TrCmaux (T, r ) , un plongement simple Go relativement B (T, r ) , oh toute palme a non incidente 51 r est telle que Int"(T(a)) = V + ( e ( a ) ) . Pour a incident B r, la propriCtC P1 assure Figs. l(b) et l(c).)
aa
une position analogue en r. (Voir
Lemme 1. U n plongement sans croisements Go d'un graphe G 2-connexe est relativement a tout arbre de Tre'maux (T, r ) un plongement de Tre'maux. Soient G" un plongement sans croisements et (T, r ) un arbre de TrCmaux de G. Supposons que pour une palme a non incidente a r, il existe x E Into(T(a)) avec x $ V + ( e ( a ) )Comme . x f r, il existe h E T, avec u+(h)= x. On a x E V + ( h ) et
V + ( h ) cInt"(T(a)),
puisque G" est sans croisements. Soit I'arste f E T qui rend maximal V'(f), avec la propriCtC x E V + ( f ) et
Gr2ce
a
V + ( fc ) Int"(T(a)).
h, f existe, et en vertu de la maximalit6 v - ( f )$ Int"(T(a)).
En fait v - ( f ) = v + ( a ) .On applique B f la propriCtC P2, d'oh il dCcoule l'existence d'un arc p incident a un sommet de Int"(T(a)) et B un sommet de Ext"(T(a)). p croise T ( a ) ,d'oh la contradiction. Une constquence du Lemme 1 est qu'il suffit, pour tester la planariti d'un graphe G, de considCrer la classe des plongements de TrCmaux associCs B un arbre de TrCmaux donnC, ou plus largement, des plongements simples relativement 2 un arbre de TrCmaux-il en existe en vertu de P4-lesquels se transforment aisbment en plongements de TrCmaux comme on le montre au Paragraphe 4.
3. Paires d'entrelacement Etant donnC un arbre de TrCmaux (T, r ) de G, on appelle paire d'entrelacement relative a (T, r ) toute paire de palmes disjointes { a ,p } telle que
T ( a )n T(P)# @ On note P l'ensemble des paires d'entrelacement de G relatives 2 ( T ,r ) . ( P est une rtduction significative de l'ensemble de toutes les paires d'arstes disjointes retenu par Tutte). Le choix de P est justifii par le lemme suivant.
Preuoe algibrique d u cridre de planariti de Wu-Liu
71
Lemme 2. Dans un plongement de Tre'maux toute paire d'arztes disjointes croisies est une paire d'entrelacement. Soient a et p deux aretes disjointes qui se croisent dans un plongement Go de TrCmaux relativement 9 (T, r ) . a et p sont des palmes puisque Go est simple. On suppose de plus que {a,p } $ P, ou T ( a )n T ( p )= 8. En vertu du thCorkme de Jordan, les courbes fermCes T ( a ) et T ( @sans ) arc commun par hypothkse, avec un croisement unique sur a et p, se coupent en un sommet o de G; il n'y a en effet pas de croisement dans l'arbre d'un plongement simple. Le sommet o ne peut &re incident aux deux palmes disjointes a et p, mais doit &re incident 9 l'une des deux, vu l'orientation de TrCmaux. Par exemple on a o = v ' ( a ) , et donc r 6 u ' ( p ) < u+(a).Alors pour le sommet x = u p ( @ ) on a: x E Into(T(a)) et
x$ V'(e(a)),
Go n'est donc pas un plongement de TrCmaux. D'oij la contradiction et le Lemme 2. On remarque que dans un plongement simple relativement 9 (T,r ) , non de TrCmaux, deux arCtes disjointes croisCes peuvent avoir des cycles fondamentaux avec un seul sommet commun. I1 apparait trois types de paires d'entrelacement relatives 9 un arbre de Trkmaux. Si { a ,P}E P, o'(a) et v'(p) sont nkcessairement distincts et comparables selon ( V , <). On peut poser u'(a)< v ' ( p ) et on a nkcessairement:
4.) $ T(P)
e ( P )E T ( a ) .
Comme a et p sont disjointes on ne peut pas avoir: E(a)E T (P ) & I?@)
E
T(a).
I1 reste trois cas possibles d'entrelacement relativement 9 un arbre de Trtmaux: E(a)$ T(P) & T(a), type I: type 11: E(a)E T ( P ) & ? ( P ) $ T ( a ) , type 111: E(a)$ T ( P ) & E ( p ) $ T ( a ) . (Voir Fig. 2.)
On dCfinit enfin le produit d'entrelacement de deux ensembles d'arCtes El, E 2 c E, Par El .E2={{a,p}EPIaEE,,pEE2}.
4. Vecteurs croisements de Trbmaux On donne un arbre de TrCmaux ( T ,r ) de G. On appelle vecteur croisements d'un plongement simple G' de G relativement 9 ( T ,r ) , l'ensemble X' des paires de palmes disjointes croisCes de G'.
P. Rosenstiehl
72
Type
'
Type I1
Type Ill
Fig. 2. Trois types de paires d'entrelacement.
Lemme 3. Si deux plongements simples Go et G1 relativernent h (T,r ) ne difirent que par la position de /3$ T, et si Po et p' sont telles que Int"(p"p') = V+(e)\{v-(p)}, oc e E T, et v - ( e ) = v+(p) ou v - ( e ) = u p @ ) , alors on a: X o + X'
= 6 V + ( e )*
{p},
oii le produit en second membre est le produit d'entrelacement, On dit alors que G' est obtenu B partir de Go par commutation de e et p en v-(e). (Voir Fig. 3(a).) I1 va de soi, en vertu du thCoreme de Jordan, que la commutation de e et p en u-(e) est exCcutable si en v - ( e ) il n'existe pas d'arete f~ T entre e et p. Si Go et G1 sont des plongements simples qui ne different que par p, leurs vecteurs croisements ne different, en vertu du thCoreme de Jordan, que de croisements {a,p } avec a E 6 Into(pop'), et {a,P } E P. Or I'enoncC fixe Into(pop'). L'exception de v-(p) dans I'CnoncC est 18 pour le cas e = e ( p ) ;son omission dans le cobord ne peut pas produire d'arete a disjointes de p. D'oh 1'CgalitC cherchCe. On pose SV'(e) { p }= A(e, p ) et on appelle A(e, p ) : vecteur commutation de e et p en v-(e). Un corollaire immkdiat du Lemme 3 est le suivant.
-
Lemme 4. A un plongement G' simple relativement h (T,r ) on suit associer un plongement de Trkmaux G 2 avec X 2 = X ' . En effet chaque palme p peut Stre commutCe en v'(p), dans un ordre convenable, avec toutes les arCtes e g T comprises entre p et e(@). Pour a E W ' ( e ) , T ( a )f l T ( p )= et donc {a,p } $ P. D'oh h(e, p ) = 0, et X2= X'.
Preuve alge'brique du critire de planarile' de Wu-Liu
x"+ x ' = (6, p }
{E,
P}
(b) type p : vecteur p ( e , f )
Fig. 3. Les deux types de commutation.
73
74
P. Rosenstiehl
Lemme 5. Si deux plongements simples Go et G' de G relativemenr a (T,r ) ne diffirent que par la position de e E T avec Int"(e"e') = V - ( f ) ,ou f~ T et v - ( e ) = u-(f), et par la position de routes les ar2tes p telles que f~ T ( p ) et v-(p) # u-(f) auec I n t o ( p o p l ) =V + ( e ) ,alors on a: X o + X ' = s v + ( e ) sV+(f). On dit alors que G' est obtenu B partir de Go par commutation de e et f en v - ( e ) = v-(f). (Voir Fig. 3(b).) I1 va de soi que la commutation de e et f en v - ( e ) est exCcutable si en u-(e) il n'existe pas d'ar&te g e T entre e et f. Si Go et G Ldiffkrent comme il est dit dans le lemme, les vecteurs X o et X i diffkrent de toute paire d'entrelacement { a ,p } pour laquelle e E T ( a )et f~ T(P). D'oU 1'CgalitC cherchCe. On pose s V + ( e )* SV+(f)= p ( e , f), et on appelle p ( e , f): vecteur commutation de e et f en v - ( e ) = v-(f). On considkre pour un arbre de TrCmaux ( T ,r ) de G donne, I'espace 9 ' sur GF(2) dont la base canonique est P. On appelle espace des commutations d e G relatives B (T,r ) , le sous-espace % de 9 engendr6 par le systkme g6nCrateur 2 constitud de tous les vecteurs de commutation associCs A (T,r ) :
4
A(e, p ) = s V + ( e )* { p }
pour tout e E T et tout p $ T avec e = e ( p ) ou v - ( e ) = v-(p),
p ( e , f ) = s V + ( e ) SV+(f) pour tout e, f e T avec v-(e) = v-(f).
Apparait alors un lemme spCcifique des arbres de TrCmaux qui permettra d'tnoncer le critkre de planarit6 de Wu-Liu.
Lemme 6. Toute paire d'entrelacement relative a (T, r ) appartient a deux vecteurs exactement du systime ge'ne'rateur 2 associe' ( T ,r ) . En effet, on considkre un arbre de TrCmaux (T,r ) et le systkme gCnCrateur 2 associ6. Pour une paire d'entrelacement {a,p } on a exactement, selon son type: type I:
{a,p } e A ( e ( p ) ,p ) et { a ,P}E A(e, p ) avec e E T, e E T ( a ) et v-(e) = v - ( p ) ,
type 11:
{a,p } E A ( e ( p ) ,p ) et { a ,P}E A(e, a ) avec e E T, e E T ( p ) et v-(e) = u - ( a ) ,
type 111: { a ,PIE A(e(P),P ) et { a ,P I E d e , f ) avec e, f e T, e E U a ) , f~ T ( p ) et v - ( e ) = v-(f) Le Lemme 6 suggkre la dCfinition suivante: on associe B (T,r ) le graphe H ( T , r )
Preuoe alge'brique du critire de planarite' de Wu-Liu
75
dont les sommets sont les vecteurs 8 E 2 , et dont les ar&tes sont les paires d'entrelacement {a,p}: {a,p } est incidente B O1 et O2 si {a,P}E el et {a,P}E 02. I1 y a une correspondance biunivoque canonique entre les vecteurs de % et les cocycles de H ( T , r ) .
5. Les criteres de planarite
La condition suffisante du premier thtorbme de ce paragraphe pourrait Stre dCmontrCe comme un corollaire du thCor6me de Tutte [9,3]. Toutefois, pour nous affranchir du r61e des configurations exclues de Kuratowski nous donnons une dkmonstration directe qui tire parti de la configuration de TrCmaux.
Theoreme 1. Soit pour un arbre de Tre'maux (T, r ) un plongement de Trdmaux GO de G. G est planaire si et seulement si le vecteur croisements X" de Go appartient a l'espace % des commutations de G relatives h ( T ,r ) . Soient un arbre de TrCmaux ( T ,r ) de G, le systbme gCnCrateur 2 et I'espace % associts. On donne un plongement de TrCmaux Go de G relativement B ( T ,r ) .
Condition ne'cessaire. S'il existe un plongement GI de G sans croisements, on peut construire une suite de commutations de type A et p assocites B ( T ,r ) qui transforment G" en G', oh chaque plongement intermidiaire construit est simple relativement h (T, r ) . Les modifications de croisements opCrCes sont donc donntes par les vecteurs A(e, p ) des commutations A o p tr te s et les vecteurs p ( e , f) des commutations p opCrCes. On note 8~ % leur somme. En vertu des Lemmes 3 et 5 , on a: x O + x l =8. O r X ' = 0, donc X"E%. Condition suffisante. Si pour le plongement GO de G on a X"E%,X" peut &tre exprim6 comme une somme de vecteurs A(e, p ) et p ( e , f ) du systkme gCnCrateur 2. On note 2(X") I'ensemble des vecteurs de cette somme. Si les commutations A et p assocites aux vecteurs de Z(X") sont extcutables dans un certain ordre sur G", elles transforment G" en un plongement GI relativement B (T,r ) de vecteur croisements X', et e n vertu des Lemmes 3 et 5 on a: x " + x l = x O ou
XL=0.
O r en vertu du Lemme 4, G' peut Stre ramen6 B un plongement de TrCmaux G2 avec X2 = X I , donc X2 = 0. Du Lemme 2 et de la proprittt P3 il dCcoule alors que G est planaire. I1 reste B prouver que Ies commutations associCes a C(X") sont effectivement exkcutables.
P. Rosenstiehl
76
I (a) a ne croise pas
1 dans C"
P9 I
1
(b) a croise p dans Go
Fig. 4. Satisfaction des commutations de
Z(X")a la rkgle (i).
Les commutations, pour &tre exicutables de proche en proche, doivent satisfaire aux deux rkgles de cohirence suivantes: (i) Si en un sommet u deux ar&tes a, b E T sont commuttes et si en u I'ar&te c E T est issue de u et placie entre a et b, alors c doit &re commutte en u avec a, ou b, ou les deux. (Voir Fig. 4.) (ii) Si en un sommet u deux ar&tes a $ T et b E T sont commuties et si en u l'arete c E T est issue de u et placie entre a et b, alors c doit &trecommutte en u avec a, ou b, ou les deux. Pour virifier que la rkgle (i) est toujours satisfaite par C ( X " ) ,on considkre trois ar&tesa, b, c E T issues d'un sommet u, avec c issue de v et placie entre a et b. On suppose que p(a, b ) # 0, e n d'autres termes qu'il existe a # T avec a E T ( a ) ,et p $ T avec b E T ( p ) oh u'(a) # u'(p). Et on pose u + ( a )
Preuoe afgdbrique du critire de planaritd de Wu-Liu
77
oii v+(y) # ~ ' ( aet) v + ( y ) # v c ( p ) .Deux cas se prCsentent alors: (a) a ne croise pas p dans G o : { ap, } # X O . Dans ce cas, puisque a commute avec b selon C(x"), p doit commuter avec e ( p ) ; cela en vertu du Lemme 6 appliquC B {a,p}. - Si v+(y)
v'(p), ou bien y croise a et pas p dans Go, ou bien y croise p et pas a ; il s'ensuit que pour celle croisCe, selon C ( X o ) y commute avec e ( r ) , et pour celle non croisCe c doit commuter avec a ou b selon le cas; cela en vertu du Lemme 6 appliquC B {a. y} et {p, y}. (b) a croise p dans G o : { aP}E , X o . Dans ce cas, puisque a commute avec b, selon C(x"), p ne doit pas commuter avec e ( p ) ; cela en vertu du Lemme 6 appliquC a {a, p}. - Si v + ( y ) < u + ( p ) , y croise p dans Go.Alors, puisque p ne commute pas avec e ( p ) selon C(x"), c doit commuter avec b ; cela en vertu du Lemme 6 appliquC a {p, y}. - Si u + ( y ) > v ' ( p ) , ou bien y croise a et pas p dans Go, ou bien y croise p et pas a. 11 s'ensuit que pour celle croiske, y commute avec e ( y ) , selon C(x"), et pour celle non croiste c doit commuter avec a ou b selon le cas; cela en vertu du Lemme 6 appliquC B {a,p } et {p, y}. Pour vCrifier que la rbgle (ii) est toujours satisfaite par C ( X o ) ,on reprend le meme raisonnement que celui fait pour la rbgle (i) en remplaGant simplement partout a par a. Donc dans tous les cas, l'argument du Lemme 6 impose aux commutations de C ( X o ) d'6tre cohCrentes. Le thkorbme est dCmontrC. Nous retrouvons maintenant le critbre de planarit6 de Wu-Liu [4, 10-121.
ThkorBme 2. Soient un arbre de Trkmaux (T, r ) et un plongement de Trkmaux Go de G relativement h (T, r ) . G est planaire si et seulement si le vecteur croisements X" de Go est un cocycle de H ( T , r ) . Le ThCorbme 2 est la traduction du ThCorbme 1 selon la dCfinition de H(T, r ) donnCe au Paragraphe 4.
Nota. Les considCrations algorithmiques sont traitkes dans un autre article en collaboration. Remerciement Nous remercions Lucas qui nous a rkvtlk les arbres de TrCmaux, le Professeur L.K. Hua qui nous a rtvtlt les derniers travaux des mathkmaticiens chinois, et notre collbgue H. de Fraysseix pour ses remarques prkcieuses.
78
P. Rosenstiehl
Bibliographie [ l ] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970). [2] J.C. Fournier, Proprittts combinatoires et algtbriques des graphes planaires, JournCes de Combinatoire et Informatique, C.N.R.S., Universitt d e Bordeaux I (1975) 153-179. [3] R.B. Levow, On Tutte’s algebraic approach to the theory of crossing numbers, Proc. 3rd S-E Conf. Combinatorics. Graph Theory and Computing (1972) 3 15-324. [4] Y. Liu, Acta Math. Appl. Sinica 1 (1978) 321-329. [ S ] E. Lucas, RCcrCations MathCmatiques (Paris, Blanchard, 1960). [6] 0. Ore, The Four-Color Problem (Academic Press, New York, 1967). [7] P. Rosenstiehl, Labyrinthologie mathtmatique, Math. Sci. Hum. 33 (1971) 5-32. [8] R. Tarjan, Depth-first-search and linear graph algorithm, SIAM J. Comput. (2) (1972) 146-160. [9] W.T. Tutte, Toward a theory of crossing numbers, J. Combinatorial Theory 8 (1970) 45-53. [ l o ] W. Wu, A Theory of Imbedding, Immersion and Isotopy of Polytopes in a Euclidean Space (Science Press, Peking, 1965). [ l l ] W. Wu, Shuxue D e Shijian Yu Renshi I (1973) 20-40. [I21 W. Wu, Kexue Tongbao 5 (1974) 226-228.
Annals of Discrete Mathematics 9 (1980) 79-82 @ North-Holland Publishing Company
REDISCOVERY AND ALTERNATE PROOF OF GAUSS’S IDENTITY Roberta S. WENOCUR Drexel Uniuersity, Philadelphia, PA I91 04, U.S.A.
0. Introduction Gauss’s identity,
where a and b are integers satisfying b - 1 > a 3 0, or, more generally,
for b - 1 > a 2 0 , a and b not necessarily integers, has been previously proved by direct means. In this short paper, we first examine such a derivation, then present an alternate proof of (0.1) based upon probabilistic considerations. In fact, when considering waiting times related to the order statistics of a previous random sample of fixed size, (0.1) emerges as a natural consequence. Note that sums (0.1) and (0.2) are special cases of the hypergeometric series F(a, c; b; 1)= *Fl(a,c ; b ; l), and can be evaluated by means of Gauss’s summation theorem for this series, with c = 1. (See, for example, Slater [4,pp. 27-28]).
1. Direct derivation of Gauss’s identity A sum of the form
where a and b are integers satisfying b - 1 > a 2 0, can be determined as follows, 79
R.S. Wenocur
80
with the Beta function denoted by B ( . ,
1
-
1
(b-a-l)! -
+):
rl
J
~ " ( 1 u- ) ~ - ~ -du '
0
a! B(a+l, b-a-1) (b-a-l)! (b-a-l)(b-l)!
Direct generalization of the above calculation yields
r ( a + k ) - T ( a + 1) k = 1 T ( b+ k ) - ( b - a - l ) r ( b ) ' for b - 1 > a 30, and where a and b are not necessarily integers. Closed forms analogous to (1.2) and (1.3) follow immediately for finite sums:
or, more generally:
Of course, a sum of the type
k=t
( b+ k ) ! '
(1.6)
or, more generally, of the form
i r ( a ++ k )
k=t
r(b k ) '
can be reduced to a sum of type (1.1) or (1.3) for T = a,to a sum of form (1.4) or (1.5), for T < m . Sums of types (l.l),(1.4) and (1.6) are prevalent in the theory of probability and statistics. As an example, consider the case of an inverse P6lya distribution (cf. [1, Sections 4.4 and 4.51); when calculating associated probabilities and moments, such sums appear. 2. Alternate proof
Let
Rediscouery and alternate proof of Gauss's identity
81
be a random sample from a population with continuous distribution function F. Let YN,;be the jth order statistic of this random sample, where we agree to count the rank j from lowest to highest, so that
YN.]= min{X,, X,, . . . , XN} and
YN,N = max{X,, X,, . . . , X N } . Let WN,,be the number of future independent trials, from the same distribution, needed to exceed YN,;for the first time (where we count the trial at which the exceedance occurs). Then
- N!(N-j+l)(k+j-2)! -
( j - l)!(N+k)!
To determine (2.1),we may employ the distribution function of YN,;, properties of the Beta function, and the fact that WN,;,conditional on YN,,,is a geometric random variable with parameter ( 1 - F( Y,,,;)).Another method exploits appropriate urn models developed by Sarkadi [3], Morgenstern [2], and Wenocur [5,7] to describe the behavior of order statistics. In particular, P( WN,N = k ) =
N
( k + N - l ) ( k +N)
Although
we have m
hence, the probability is 1 that the N-year maximum will be exceeded, although we expect to wait infinitely long for this to occur. The analysis of WN.,, is related, but not identical, to the theory of record times (see [6]). Since the probability is 1 that the N-year maximum will eventually be exceeded, it follows immediately that any order statistic YN,;, j = 1, 2, . . . , N, will be exceeded with probability 1. That is, + l)(k +j-2)! c N ! ( N - j1)!(N+ =1 k)!
k=l
(j-
Setting a = j - 1, b = N + 1, and noting that the restriction 1 S j S N is equivalent to b - 1 > a 2 0, (0.1) is established as an immediate consequence of (2.2).
82
R.S. Wenocur
Acknowledgments Many thanks to the referee for a valuable remark, to Dr. S. Zietz for his guidance through the stacks in Drexel’s library; to Professor J. Galambos and to Professor S. Kotz for their interest in alternate proofs.
References [ 11 N.L. Johnson and S. Kotz, Urn Models and their Application (John Wiley and Sons, New York. 1977). [2] D. Morgenstern. Uberschreitungswahrscheinlichkeiten.das Polyasche Urnenmodel und ein Wartezeitproblem bei Urnenziehungen, Math-Phys. Semest., Gottingen 19 (2) (1972) 2 13-215. [3] K. Sarkadi, On the distribution of the number of exceedances, Ann. Math. Statist. 28 (1957) 1021-1023. [4] L.J. Slater. Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966). [ 5 ] R.S. Wenocur. Waiting times and return periods related to order statistics: an application of urn models, to appear in Statistical Distributions in Scientific Work (1980). [6] R.S. Wenocur. Waiting times and return periods to exceed the maximum of previous sample, to appear in Statistical Distributions in Scientific Work (1980). [7] R.S. Wenocur, Waiting times and return periods related to order statistics, Thesis, Temple University, Philadelphia, PA (1979).
Annals of Discrete Mathematics 9 (1980) 83-85 @ North-Holland Publishing Company
DECOMPOSITION D’UN GRA PHE EN CYCLES ET CHAINES D. BRESSON Ecole des Hautes Etudes en Sciences Sociales, 54 b o d . Raspail, 75270 Paris VI, France Pour prouver que le carrt de tout graphe 2-connexe est hamiltonien, H. Fleischner [2] a montrt I’existence pour tout graphe 2-connexe d’un recouvrement connexe de ses sommets par des cycles et des chaines, tous arCte-disjoints, les chaines e‘tant de plus entre elles sommetdisjointes. Nous proposons ici une dtmonstration constructive de ce thtoreme baste sur l’algebre des cycles modulo 2, et les proprittts des arbres de Trtmeaux d’un graphe.
1. Introduction La terminologie adoptCe est celle de [l]. On considgre comme allant de soi les dkfinitions Cltmentaires des graphes non-orient& G = (V, E ) et orient& G = (V, A) o i ~V est l’ensemble des sommets, E l’ensemble des arCtes, A l’ensemble des arcs. Cycles (i.e. polygones) et chaines (i.e. chaines ClCmentaires) sont ici pris en tant qu’ensembles d’arCtes et les chemins en tant qu’ensembles d’arcs. Tout ensemble d’arcs pourra Ctre assimilt a l’ensemble d’arCtes correspondant. Une chaine, comme une arCte, est incidente a une paire de sommets; un arc a (resp. un chemin II.) est incident 2 un couple de sommets ( a ( a ) ,@(a))(resp. ( a ( p ) ,p ( p ) ) ) .Un sommet u est incident 2 F c E, si et seulement si v est incident une artte de F. R c E est un recouvrernent des sommets de G, si tout sommet de G est incident a R. F c E est connexe si, pour toute paire de sommets x et y incidents a F, F inclut une chaine incidente a x et y. On considkre connus l’algkbre des cycles modulo 2 et le fait qu’un cycle algCbrique est toujours une sornrne de cycles-polygones disjoints. De mtme seront considtrkes comme connues les dCfinitions d’un arbre maximal T de G, des cycles fondamentaux y ( e ) associCs a tout e & T. G est dit 2-connexe si par tout r et s de V, il passe un cycle polygone. On considkre aussi comme allant de soi, I’ordre partiel des sommets ou des arcs de T, induit par l’enracinement de T en un sommet r. On note alors T l’ensemble des arttes orientCes de T (ou arcs). On appelle arbre de Trkrneaux T (voir [4, 5,3]), un arbre maximal T de G, enracinC en un sommet r, et tel que toute arCte e & T joint deux sommets comparables pour I’ordre partiel induit par cet arbre. Les arCtes e& T sont orientkes par dCfinition de a ( e ) a @ ( e ) de sorte que @ ( e ) < a ( e ) si r est pris 83
84
D . Bresson
comme premier sommet dans l’ordre sur T. Ces arcs sont alors appelCs palmes de T. On note deux lemmes ClCmentaires concernant les arbres de TrCmeaux:
Lemme 1 Si G est 2-connexe, la racine r de Test incidente a a u moins une palme. Lemme 2. Si G est 2-connexe, alors il existe un seul arc a E T avec a ( a ) = r. 2.TheorBme TheorBme. Tout graphe 2-connexe admet un recouvrement connexe de ses sommets par des cycles et des chaines, tous arite-disjoints, les chaines &ant de plus entre elles sommet-disjointes. On considkre un graphe 2-connexe G a n sommets et, sur G, un arbre de TrCmeaux T de racine r. On note P l’ensemble des palmes de T et y(p) le cycle associt dans T tout p E P. On se donne alors dans T une numkrotation des arcs {ai;i E [l, n - 13) = T, compatible avec l’ordre partiel de l’arbre enracinC en r. Le recouvrement connexe de G annonck, R, est constituC des deux ensembles disjoints suivants: - du cycle algCbrique T = CpePy(p), - des chaines sommet-disjointes dont la rCunion est C = A,-, \ T oh A,-,, est I’ensemble d’arktes de G dCfini par l’algorithme a n-1 Ctapes Ctabli ci-dessous, et dCpend de I’order total sur T donnC ci-dessus.
Algorithme. Initialement on pose: A,, = (a, So = (d;
- A 1’Ctape i (i E [l; n - 13 on considkre ai: (i) si ai E A,_, alors Si = Si-l, Ai = Ai-,, (ii) si ai E r \ A i - , alors Si = Si-l, Ai = + a,, (iii) si a i E ( T \ T ) \ A i - l alors Si= S i - l + a i , Ai = A i P l + p i oh pi est le chemin allant de /3(ai) /3(pi) le long d’un des cycles fondamentaux y(pi) arbitrairement choisi parmi ceux passant par ai.
Remarque. Tout ai, avec j s i appartient soit a Si(supprimk), soit a Ai (accepti). Preuve. L’algorithme produit effectivement A,-, en n - 1 Ctapes, car tout ai appartient a au moins un cycle fondamental puisque G est 2-connexe. On prouve que r +C est le recouvrement annoncC en trois Ctapes: (1) T + C est un recouvrement des sommets de G. On a: A , - , c T + C par dCfinition de C. Si A,-, est un recouvrement, il en est a fortiori d e mkme pour T + C. On montre que A,-, est un recouvrement en prouvant que tout sommet v de G est incident 2 An-l. - Soit v = r alors, ou bien a, E A , et donc r est incident a A,-,, ou bien a, E S, et alors r est incident a la palme p, et, comme P , E A , , r est incident a An-l.
Decomposition d’un graphe en cycles et chaines
85
- Soit v f r, alors on considkre I’arc unique ai E T tel que p(ai) = ui alors, ou bien a, E Ai et donc v est incident a An-l, ou bien v est incident a pi et, comme pi c A,, u est incident a An_1. ( 2 ) r + C est connexe. Si An-l est connexe, il en est a fortiori de m&mede I-+ C. On montre que AnPlest connexe par rbcurrence sur i E [ l ; n - 11. On suppose AiP1connexe. On considkre les 3 cas de I’algorithme: - Dans le cas (i), Ai = Ai-,, donc Ai est connexe; - Dans le cas (ii), Ai = Ai-l a,. On considbre alors pour i > 1 (cf. Lemme 2), l’unique arc a, tel que p(ai>=a ( a i ) . Comme j = Z i - 1 par dCfinition de la numCrotation des arcs, on a (voir remarque de l’algorithme) soit ai € A i - ,et alors P(a,) est incident a et a ai; Ai est connexe. soit ai E Si+l,et alors p(ai) est incident h pi et donc a Ai+,ainsi qu’a a i ;Ai est connexe. - Dans le cas (iii), Ai = AiP1+ pi. On considkre pi soit p ( p i )= r, alors r est incident h Ai et a p,, donc Ai est connexe. * soit P(pi) # r, et alors pour I’arc ai E T avec p ( a i )= p(pi), on a j = z i 1 et on raisonne comme au cas prCcCdent oh ai est maintenant pi.
+
(3) C est la re‘union de chaines sommet-disjointes. On considkre I’ensemble des chemins hi = pi \pi de T pour i E [ l ; n - 11. On a C c U hi par definition de C. Si UAi est une for6t de chemins, il en est a fortiori de m6me pour C. C sera la rbunion de chaines sommet-disjointes annoncbe. On montre que la for& U hi est une for&t de chemins en considbrant la bijection qui associe a tout hi existant, l’arc ai qui a CtC supprim6 en lui donnant naissance. I1 s’ensuit que, pour if j , on a a ( h i )# a ( & ) .Or les hi Ctant des chemins maximaux de leur forct, les a ( & )sont racines de la for&. Donc les racines de la for6t sont en bijection avec les hi, lesquels sont ainsi sommet-disjoints.
Remerciement Nous remercions H. de Fraysseix et P. Rosenstiehl pour leurs prCcieux conseils.
Bibliographie [ l ] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1973). [2] H. Fleischner, On spanning subgraphs of a connected bridgeless graph and their application to DT-graphs, J. Combin. Theory 16 (B) (1974) 17-28. [3] J. Hopcroft et R. Tarjan. Efficient planarity testing. J. Assoc. Comput. Mach. 21 (4) (October 1974) 549-568. [4] E. Lucas, Thiorie des nombres (Paris, 1891). [S] P. Rosenstiehl, Labyrinthologie mathimatique, Math. Sci. Humaines 33 (197 1) 5-32.
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Annals of Discrete Mathematics 9 (1980) 87-91 @ North-Holland Publishing Company
THEORIE DE LA MEDIANE DANS LES TREILLIS DISTRIBUTIFS FINIS ET APPLICATIONS B. MONJARDET Uniuersiti de Paris V et Centre de Mathkrnatique Sociale, 54 boulevard Raspail. 7.5270 Paris Cedex Oh. France We present broad generalizations of the notion of median in finite distributive lattices. which can be used to characterize these lattices. We point out applications of this theory of generalized medians in domains as social choice theory or cluster analysis.
1. Introduction
I1 est bien connu que dans un treillis distributif tout triplet d’C1Cments (xI, x2, x3) admet une mCdiane
m (Xi,
X2. X3)
= (XI A Xa) V ( X 2 A X3) V
X I ) = ( X i V X2)A ( X 2 V Xg)A ( X 3 V X i )
et que cette propriCtC est caractkristique des treillis distributifs. Soit maintenant u une valuation positive d’un treillis distributif (par exemple, une fonction de rang) et d la distance associCe en posant d(x, y ) = V ( X v y) - U(X A y ) .
Un exercice dans Birkoff [ 5 , p. 431, montre que la mCdiane de (xl, x2, x3) est aussi I’CICment du treillis minimisant la somme C:=’=,d ( t , xi). Dans un memoire, malheureusement non publik [l], M. Barbut a donne, dans le cas de parties d’un treillis distributif, d’importantes gCnCralisations de ces rCsultats. On peut les Ccrire, pour le cas de n-uples d’CICments d’un treillis distributif sous la forme suivante: (1) Pour tout n-uple X = (xi)icr d’un treillis distributif on peut dCfinir un intervalle median dont les ClCments seront appelCs les mkdianes de X. Dans le cas (I1= 2p + 1, la mCdiane est unique et s’Ccrit suivant les deux expressions duales:
m(~)=
v
KcI IKI=p+l
( iAe K xi)=
A
KcI IKI=p+l
( v xi) ieK
(2) Pour une distance d associCe h une valuation u sur le treillis distributif, toute mCdiane du n-uple X minimise la quantitC d ( t , xi). (3) L‘existance de mCdianes (gCnCralistes) est caractkristique des treillis distributifs. Les rCsultats (1) et (3) peuvent eux-m2mes 2tre gCnCralisCs en utilisant la notion 87
B.Monjardel
88
de famille de parties (ou hypergraphe) “transversale” d’une famille de Sperner (cf., par exemple, [4] et [8]). I1 existe en effet un lien Ctroit entre la notion de famille transversale et celle de treillis distributif, lien explicit6 dans ce texte par la Proposition 1 et le ThCorkme 3. Dans le cas particulier oh le treillis distributif est un ensemble totalement ordonnC on retrouve ainsi des rCsultats d’EdmondsFulkerson, sur les systkmes de blocage [6, 91. La thCorie de la mCdiane gCnCralisCe dans les treillis distributifs fournit aussi une base B des methodes de rCsumC de donnCes relationnelles (les relations Ctant des ClCments d’un treillis boolken), qui apparaissent en thCorie du choix social (ordres ou prtordres totaux), en taxonomie mathkmatique (Cquivalences), en analyse multicritkres, etc. . . N.B. L’existence d’une mCdiane ternaire a CtC gCnCralisCe B d’autres structures que les treillis distributifs (cf., par exemple, Sholander [12] et Mulder et Schrijver [101).
2. Notations et preliminaires Dans tout le texte les ensembles considCr6s sont finis. On note D un treillis, s sa relation d’ordre, A et v ses opirations infimum et supremum. Si x < y on note [x,y] I’intervalle form6 des ClCments z tels que x < z < y . On rappelle qu’un treillis est distributif s’il verifie l’une des trois identitb Cquivalentes suivantes: x A (y V Z ) = (X (X V y ) A (X
A Y)V(XA
z).
(XVZ) = X V ( y A 2 ) .
v y ) (X ~ v Z ) A (z v y ) = (X A y)v (X A z) v (z A y ) .
Soit I = { l , . . , i , . . . , k , . . . , n} un ensemble B n CICments. Une famille de Sperner sur I est un ensemble 9 de parties de I, deux B deux incomparables pour l’inclusion. Une partie T de I est une transversale de 9 si elle a u n e intersection non vide avec toute partie appartenant B S.On note Tr S,la famille de Sperner, formCe de toutes les transversales minimales de 9. (Par exemple, si 9 est la famille de toutes les parties B k CICments de I, Tr 9 est la famille de toutes les parties B n - k + 1 Cl6ments de I). Un couple (9, Tr 9) a CtC appel6 systkme de blocage par Edmonds et Fulkerson [6]. Dans ce texte, now considkrons des n-uples (x,, . . . , x i , . . . , x,) d’CICments d’un treillis, index& par les ClCments de I. Nous les notons ou X.
Proposition 1. Soient un n-uple d’e‘le‘ments d’un treillis distributif D et 9 une famille de Sperner sur I ; on a :
v
K s 4
( iAs K xi)= T sAT r S ( iVs T xi).
N.B. Cette identit6 de distributivitk est plus commode -et plus Cconomique que la classique identit6 de distributivitk gCnCralisCe [ 5 ] .
89
ThCorie de la rntdiune dans fes treillis distributifs finis
3. Miidiane generalisee dans un treillis distributif n-uple d’C1Cments d’un treillis D quelcon-
Definition algebrique. Soit X = que. Pour k E I, on pose ak=
A (.V i t K Xi).
KcI IK(=k
bk=
V ( it-K A Xi).
KcI IKI=k
Dans un treillis distributif, on appelle: associke h X, C(X) le n-uple (al = b, C * =S ( S , - k + l = bk G . . * - intervalle mkdian de X, M(X), l’intervalle [a[(,+2),21, aC(n+2),211; - mkdiane de X, m(X), tout ClCment de l’intervalle mCdian de X. Par exemple, si n = 2p + 1, la mkdiane de X est unique et s’kcrit:
-
- chaine
a,
= bl);
Remarque. Dans un treillis distributif 0, les ClCments a k (ou b k ) peuvent aussi 6tre dCfinis comme intersection ensembliste d’intervalles ou a partir des supirreductibles de D. Par exemple, si n = 2p + 1 , m(X) est l’intersection des intervalles [ r \ i c K xi, V i t Kxi] pour IKI = p + 1, et est le supremum des sup-irrkductibles de D infCrieurs 51 au moins p + 1 ClCments de X (procedure “majoritaire”). Definition metrique. Soit u une valuation positive [2, 51 sur le treillis distributif D et d la distance associCe par d ( x , y ) = u ( x v y ) - v ( x A y). On dCfinit un kloignement entre t ClCment de D et X, n-uple en posant: n
A(t, X) =
1 d(t, xi). i=l
Theoreme 1. Soit X un n-uple, et C(X) la chaine associke; on a A(?,X) = A(t, C(X)).
Theoreme 2. U n klkment t de D minimise l’iloignement A(t, X ) au n-uple X, si et seulement si t est une mkdiane de X. Pour n impair, un tel klkrnent est unique.
90
B . Monjardet
4. Caracterisation des heillis distributifs
Theoreme 3. Soient D un treillis, 9 une famille de Sperner sur I = { 1, . . . , i, . . . , n} d ’ t l h e n t s de D, on a ( 1 3 13 2, n 5 3). Si pour tout n-uple
le treillis D est distributif.
Corollaire. Si D est un treillis oir tout n-uple ( n impair fix4 2 3 ) d’tltments udmet une mtdiane (i.e. l’tgalitt (1) est vkrifide) D est distributif. Remarque. Prenons I = {1,2,3} et pour families de Sperner les familles 9, = = (12, 13}, g3 = (12, 13,23}. Le thkorbme redonne alors les trois (1,23}, 9* caractkrisations classiques des treillis distributifs rappellCes au paragraphe 2.
5. Applications Une “statistique ordinale” est une application d’un ensemble I d’ “individus” dans un ensemble ordonnC quelconque D. Si D est un treillis distributif, les rCsultats prkcbdents fournissent une thCorie pour rCsumer de telles statistiques. Si D est un ensemble totalement ordonnk on retrouve la notion habituelle de mkdiane (les “quantiles” usuels correspondant aux ClCments u k ) . Un cas particulikrement intkressant est celui oh D est le treillis boolken P(E2) des relations binaires sur un ensemble E. E n effet beaucoup de donnkes de prkfkrences, de classements se prksentent comme des relations binaires d’un certain type: prkordres ou ordres totaux, tournois, quasi-ordres, Cquivalences, etc. Le problbme de rksumer (d’agrkger) de telles donnkes est classique. Par exemple, lorsque les donnkes sont des ordres totaux on doit 2 Condorcet la mkthode dite des comparaisons par paires, consistant B retenir pour chaque paire de “candidats” B comparer, celui prCfkrC par une majorit6 de votants. Cette m6thode revient en fait a calculer dans le treillis des relations binaires, la relation mkdiane des ordres totaux initiaux. Le fait que cette relation mCdiane n’est pas nkcessairement un ordre total est l’effet Condorcet bien connu et conduit a la recherche d’ordres “mkdians”. On retrouve la m2me situation en taxonomie mathkmatique lorsqu’on cherche un “bon” rCsumC de partitions d’un ensemble E par une partition “centrale” [ll]. Nous renvoyons B BarthklCmy et Monjardet [3] et Marcotorchino et Michaud [7], pour une vue d’ensemble sur ces problbmes, l’ouvrage des seconds dCveloppant particulibrement des algorithmes de rksolution.
The'orie de la me'diane dans les treillis distributifs finis
91
Bibliographic [ 11 M. Barbut, Mtdiane, distributivitt, Cloignement, publication du Centre de Mathtmatique Sociale, E.P.H.E., Paris (1961), et Math. Sci. Hum. 70 (1980) 5-31. [2] M. Barbut et B. Monjardet, Ordre et classification: algtbre et combinatoire, 2 tomes (Hachette, Paris, 1971). [3] J.P. Barthelemy et B. Monjardet, The median procedure in cluster analysis and social choice theory, Math. Social Sci. 1 (3) (1981). [4] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1070). [S] G. Birkhoff, Lattice Theory (AMS, Providence, NJ, 1967). [6] J. Edmonds et D.R. Fulkerson, Bottleneck extrema, J. Combin. Theory 8 (1970) 299-306. [7] J.F. Marcotorchino et P. Michaud, Optimisation en analyse ordinale des donntes (Masson, Paris, 1979). [8] B. Monjardet, Elements ipsoduaux du treillis distributif libre et familles de Sperner ipsotransversales, J. Combin. Theory 19 (1975) 160-176. [9] B. Monjardet, Systkmes de blocage, Cahiers du C.E.R.O. 17 (2-3-4) (1975) 293-294. [lo] H. M. Mulder et A. Schrijver, Median graphs and Helly hypergraphs, Discrete Math. 25 (1979) 41-50. [ 1 11 S. Regnier, Sur quelques aspects mathtmatiques des probltmes de classification automatique, I.C.C. Bull. Rome 4 (1965). [12] M. Sholander, Trees, lattices, order and betweenness. Roc. Am. Math. SOC.3 (1952) 369-381. [13] J.B. Miller, Introduction to a theory of coups, Algebra Universalis 9 (3) (1979) 346-370.
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Annals of Discrete Mathematics 9 (1980) 93-101 @ North-Holland Publishing Company
GRAPHES NOYAU-PARFAITS P. DUCHET C.N.R.S., 15 Quai Anatole, Paris, France A directed graph G is said to be kernel-perfect if every G-subgraph possesses a kernel (= a stable dominating subset). Graphs without circuits, graphs without odd circuits are known to be kernel-perfect (Von Neumann and Morgenstern, and Richardson), and so are symmetric or transitive graphs. I present here the current state of research in kernel theory; the most significant results are the proofs of kernel-perfectness of G in each of the following cases:
Theorem 2.2. G is a right- (or left-)pretransitiue digraph Theorem 3.3. (see [5]). Euery odd circuit of G (not elementary in general) u , . . . u z p + l u , possesses two crossing short chords (that means there exist two arcs of G of the form (vq, u,,,) and (u,,,, uq+&
Theorem 4.2. Euery circuit of G possesses at least one symmetrical arc Theorem 4.3. Euery odd circuit of G possesses at least two symmetrical arcs. Theorems 3.3 and 4.3 are particuliar cases of an interesting conjecture proposed by H. Meyniel. Somes curious kernel-critical graphs are exhibited.
0. Quelques precisions terminologiques Les graphes considCrCs ici sont orient& sans boucles. On posera G = ( X , U ) ; nous noterons u- l’arc (b, a ) opposi h un arc u = ( a , b ) ; pour T c U, on note T- = { u - : u E T } ; G - dksigne le graphe ( X , V-); un arc u de G est dit syrnktrique si u- est aussi un arc de G. Un parcours du graphe G est une sCquence u1 * u, d’arcs de G oh I’extremitC initiale de coincide avec 1’extrCmitC finale de ui, pour i = 1, . . . , rn - 1. Le parcours est dit fermt si I’extrCmitC finale de u, coincide avec 1’extrCmitC initiale de u,. Les y sont les arcs du parcours et les extrkmitb des ui sont les sommets du parcours. Un chernin est un parcours oh tous les arcs sont diffkrents, un circuit est un chemin fermC. Rappelons enfin les notations classiques:
TG(a)={b:(a,b)E U } et rG(A)= U I‘G(a), a€A
&(b) = { a : (a, b ) E U } et
T&B) = U I‘,(b). bEB
93
P. Duchet
94
1. Graphes noyau-parfaits
1.1. Le concept de noyau fut introduit en 1944 par Von Neumann et Morgenstern comme ‘‘solution’’ d’un jeu 21 n joueurs et B somme nulle. 11s indiqukrent notamment que tout graphe sans circuit posskde un noyau unique. Une solution est, dans leur sens, un ensemble d’imputations (= paiements du jeu) qui prbsente des propriktks de stabilitk interne et externe. En langage de graphes , un noyau du graphe G = (X, U ) est une partie N de X telle que:
T , ( N ) n N = P, (stabilitC “interne”), & ( N ) UN
=X
(1.1.1)
(stabilitk “externe” ou absorption ou “dominating” subset). (1.1.2)
1.2. Berge proposait (dans [2]) de caractkriser les graphes noyaux-critiques, graphes sans noyaux tels que l’klimination d’un sommet quelconque produise un graphe avec noyau; il semble irrkaliste de chercher une rkponse gCnCrale a ce probkme; les circuits impairs, les graphes des Figs. 1 et 2 donnet des exemples de graphes noyau-critiques. I1 semble plus abordable de chercher B reconnaltre les graphes minimaux sans noyau, graphes sans noyau et dont tout sous-graphe induit strict a un noyau. I1 est en effet remarquable que les conditions suffisantes connues, assurant l’existence d’un noyau dans un graphe assurent e n fait la noyau-perfection du graphe, c’est a dire l’existence d’un noyau pour tout sous-graphe induit:
Theoreme 1.3. Un graphe syme‘trique est noyau-parfait. Theoreme 1.4. Un graphe transitif est noyau-parfait. 0
2
2
5
5
71.2 arcs (x, y): y - x ~ { 1 , 2 }(mod7)
71,2,s arcs (x. y): y - x ~ { 1 , 2 , 5 } ( m o d 7) Fig. 1.
’
1 1.2.4 arcs (x, y): y - x € { 1 , 2 , 4 } (mod 11)
Graphes noyau-parfaits
95
arcs (x, y ) : arcs (x. y ) : y - x E { 1,2} (mod 7)
arcs (x’, y’): y - x E {O, 1,2) (mod 7) arcs ( y ’ , x): y - x = l (mod7) y - x E {3,4,5,6) (mod 7) Fig. 2.
Ththreme 1.5 (Von Neumann et Morgenstern [8]). Un graphe sans circuit est noyau -parfait. Theoreme 1.6 (Richardson [7]). Un graphe sans circuit impair est noyau-parfait. (Voir la preuve de Von Neumann in [2, p. 2991.) Une premikre formulation englobant les ThCorkmes 1.3 i 1.6 fut le resultat suivant, facile 6 Ctablir soit en utilisant la preuve originelle du thCorkme de Richardson ([l, p. 299-300; ou 71) soit par une procCdure d’orientation dont on verra plus loin un exemple plus poussC:
Theoreme 1.7 (Duchet et Meyniel [ 5 ] ) . Si les arcs des circuits impairs de G sont symktriques, G est noyau-parfait. Depuis, trois id6es nouvelles ont permis d’obtenir diverses extensions des thCorkmes originels: mCthodes “rCcursives”, de “substitution”, de “rCorient at ion . ”
2. Methodes recursives Ce terme recouvre en fait des situations variCes de preuves mais dans lesquelles la caractkristique dominante est l’utilisation d’un sous-graphe ou d’un graphe partiel de G pour lequel on montre l’existence d’un noyau qui est Cgalement un noyau de G.
P. Duchet
96
2.1. Graphes prktransitifs. Un graphe G = (X, U ) est prktransitif h droite (resp. h gauche) si toute partie A de X possbde un sommet t(A)= a vkrifiant: U).
(2.1.1)
((x,a ) E U et (a, Y ) E U )3 ((x, y ) E U ou (a, x>E U ) .
(2.1.2)
((x, a ) € U et (a, Y ) E U )3 ((x, Y ) E U ou ( Y , resp.
Theoreme 2.2. U n graphe prktransitif a droite (resp. a gauche) est noyau-parfait. Preuve. Supposons G = (X, U ) prktransitif tl
droite (resp.
gauche) et posons:
=t(X), k
u
tk+l= t ( x -
i=l ({ti}
tk+l = t ( X -
i=l
u rE(ti$?
resp. k
u ({ti}u(~E(ti)\r~(ti)))).
Le sous-graphe partiel G I = ({ti}i,U‘) oh U’ est l’ensemble des arcs de G de la forme (ti,ti) avec i < j est un graphe sans circuit et on montre aiskment que le noyau de G’ est un noyau de G.
Problhme 2.3. Peut-on trouver une condition unique plus faible que chacune des conditions de prktransitivitk (a droite et a gauche) et assurant la noyauperfection? Cette question n’est toujours pas rksolue malgrC divers aff aiblissements de la prktransitivitk obtenus par Jacob [6].
2.4. Graphes orient& d’intervalles. U n exemple intkressant de graphes prktransitifs est fourni par les graphes orientks reprksentatifs d’une famille d’intervalles pointks d’une arborescence ou d’une antiarborescence. Si d = ( X , A ) est une arborescence, un intervalle de d est constituk de l’ensemble des sommets d’un chemin orient6 de d.Une famille pointke d’intervalles est un couple (9,p ) oh 9 est une famille d’intervalles non vides de d et oh p :9 .--, X est une application vkrifiant p ( I )E I pour tout intervalle I E 9. p ( 9 ) est l’ensemble des points de la famille. Deux intervalles I et J sont dits indipendants si p ( I ) $ J et p ( J )$ I. Deux points sont dits indkpendants s’ils correspondent a deux intervalles indkpendants. Le graphe orient6 reprksentatif L(9, p ) de la famille est le graphe sur 9 de la relation p ( I )E J. Theoreme 2.4.1. Si (9, p ) est une famille point6e d’intervales d’une arborescence ou d’une antiarborescence, il existe un recouvrement de p(9) form6 d’intervalles mutuellement indipendants.
Graphes noyau-parfaits
97
Th4orkme 2.4.2. De plus il existe une famille de points de ( 9 , ~mutuellement ) indipendants telle que tout intervalle de %. contienne un point de cette famille. Ces rCsultats peuvent s’Ctendre au cas des farnilles pointkes d’intervalles d’un arbre orientC. (La seule demonstration que j’ai de cette extension est trop compliquee et trop longue pour figurer ici.)
3. La conjecture de Meyniel Tout circuit impair d’un graphe noyau parfait possbde une corde, arc du graphe mais non du circuit et dont les extrCmitCs sont des sommets du circuit, la rCciproque n’Ctant pas vraie (voir ii ce propos la Conjecture 1 que je propose d a m la section des problhmes du colloque). Meyniel a proposC en 1976 au sCminaire de Paris une extension trks intkressante des thCorbmes de base sur l’existence des noyaux, que I’on peut reformuler comme suit:
Conjecture 3.1. Si tout circuit impair de G possbde deux cordes, G est noyauparfait. On peut voir que, sous l’hypothbse de la conjecture, tout circuit impair de G possbde alors deux cordes paires (une corde (a, b) d’un parcours fermC est dite de longueur k si il y a un chemin de longueur k form6 d’arcs du parcours et constituant une sous-sCquence du parcours, dont 1’extrCmitC initiale est a et I’extrCmitC terminale 6); deux cas particuliers importants de la conjecture retiennent l’attention: 3.2. Le cas oij on suppose l’existence de deux cordes de longueur 2 (cordes dites “courtes”) pour tout circuit impair n’est toujours pas rCsolu; on a toutefois les rCsultats suivants: Theoreme 3.3 (Duchet et Meyniel [ 5 ] ) . Si tout circuit impair de G possbde deux cordes courtes croisies (c’est h dire de la forme (xk, xk+J et ( x k + , , &+3) pour un circuit de sommets x1 . * * x,) alors G est noyau-parfait. L‘idte de la dCmonstration rCside dans une modification pas B pas, par substitution, du noyau d’un sous-graphe de G pour obtenir un noyau de G.
Thhreme 3.4 (Duchet [4]).Si tout circuit tldmentaire impair de G possbde deux cordes courtes non croisies (pour les circuits de longueur 25) G est noyau-parfait.
P. Duchet
98
Le Thkorbme 4.3 sera en fait une extension de celui-ci. Signalons enfin un rksultat d’un type diffkrent:
Theoreme 3.5 (Jacob [6]). Si tout circuit de longueur 3 de G a deux cordes et si tout cycle impair (non orienti) de longueur 3 5 a deux cordes non croiskes, G est noyau-parfait.
4. Methodes d’orientation des arcs symetriques 4.1. L‘idke fondamentale de la mkthode est que le graphe G posskde un noyau si et seulement si il existe un choix d’orientation des arcs symhtriques de G qui produise un graphe avec noyau. Pour S c U, une orientation partielle de S est un ensemble T c X x X tel que:
UETJU-$T.
U E T ~ U E ouS
(4.1.1)
u-ES.
Une orientation T de S est totale si:
U E S ~ U E ou T
U-ET.
(4.1.2)
Theoreme 4.2. Si tout circuit de G possl.de un arc symttrique, G est noyau-parfait. Theoreme 4.3. Si tout circuit impair de G possl.de deux arcs symttriques, G est noyau-parfait. Ces rksultats dkcoulent immkdiatement de l’idke fondamentale d’orientation et des thkorkmes de rkorientation qui suivent:
Thbreme 4.4. Soit S un ensemble d’arcs d’un graphe G = ( X , U ) tel que tout circuit de G possbde un arc de S ; alors on peut rtorienter les arcs de S de manibre a obtenir un graphe sans circuit. Preuve. Appelons orientation valide une orientation partielle T de S telle que le graphe G ( T )= ( X ,( U - S ) U T ) soit sans circuit. Une orientation valide maximale T de S est totale: en effet, si u est un arc de S avec u $ T et u-$ T., les orientations T + u et T + u- sont non valides et il existe deux circuits A et B de la forme A a1. *
. asp,
= U L Y ~. . . aa, *
B = u-0,
* * *
&,,
Pb est alors un circuit de G(T), contradiction.
Theoreme 4.5. Soit S un ensemble d’arcs d’un graphe G = ( X , U ) tel que tout
Graphes noyau-parfaits
99
circuit impair de G possbde au moins deux arcs de S ; alors on peut r6orienter les arcs de S de manibre 2( obtenir un graphe sans circuit impair.
Preuve. Remarquons tout d’abord que d’un parcours fermC impair on peut toujours extraire un circuit impair. Comme prCcCdemment, disons qu’une orientation partielle T de S est valide si le graphe G (T )= ( X , ( U - S) U T ) est sans circuit impair et appelons acceptable une orientation T de S telle que l’une au moins des orientations T + u et T + usoit valide, pour tout arc de S non orient6 par T (= pour tout arc u E S \ ( T U T-)). Soit alors T une orientation acceptable maximale de S.
Lemme. Pour tout u E S \ ( T U T-), T + u et T + u- sont valides. Supposons par exemple T + u valide et T + u- non valide. T Ctant maximale, T + u n’est pas acceptable et il existe un arc IJ E S\{u}U(TUT) tel que:
G ( T +u + u ) contient un circuit impair A, G ( T +u + v-) contient un circuit impair B. Comme T + u est valide, v figure dans A et v- dans B :
A
= val. . ’aa,
B = v-p1 * * . Pb.
En outre, comme l’une des orientation T+v ou T + v - est valide d’aprks l’acceptabilitk de T, u figure au moins une fois dans A ou B. D’autre part T + u- &ant non valide, il existe un circuit impair
c=u - y 1 * * . yc dans le graphe G ( T +u-). En remplaqant une occurence de I’arc u, et une seule, dans le parcours fermC pair a1. . . a,pI &, par le chemin pair y1 . . yc, on obtient un parcours fermC impair du graphe G ( T +u ) , ce qui contredit la validitC de T + u .
-
-
I1 suffit pout achever la preuve du thCor&me de montrer que T est une orientation totale de S. Supposons qu’au contraire il existe un arc u E S \ ( T U T-). D’aprks le lemme et le caractkre maximal de T, il existe deux arcs v et w, non nkcessairement distincts, dans S \ ( T U T- U { u } U{ u - } ) tels que les graphes G(T + u + u ) , G ( T + u + v-), G ( T +u - + w),G ( T +u - + w-)contiennent respectivement des circuits impairs de la forme:
A = va, * . * aa,
B
M=w
N=
~
*I * *
pm,
= v-pl W-VI
*
. ‘ pb,
* * *
%*
D e plus, comme T + v, T + v-, T + w et T + w - sont valides d’aprks le lemme, u figure dans A et dans B et u p figure dans M et dans N : u = ap= p,,
u- = p, = v,.
P. Duchet
100
--
pl. pq-luap+l * a, est un parcours fermk de G ( T + u) (qui ne contient qu’une occurence de u) et est donc pair. En remplaqant dans ce parcours l’unique occurrence de u par le chemin P , + ~* * p,wkI . P , - ~ , on obtient un parcours fermk impair dans le graphe G ( T + w ) , ce qui contredit la validitk de T t w. T est donc bien une orientation totale. 1
-
Ainsi que l’a montrk M. Las Vergnas les Thkorkmes 4.4 et 4.5 s’ktendent sans difficult6 aux modules unimodulaires. La dkmonstration ci-dessus fournit un algorithme e n O(1Ul) pour l’obtention d’une rkorientation convenable de S.
5. ProblBmes fermb
Le lecteur trouvera dans la section des problkmes du colloque une liste de conjectures en thkorie des noyaux. Signalons juste ici l’inexactitude de deux conjectures de Chvatal [3]: 5.1. La condition, kvidemment nkcessaire B la noyau-perfection d’un graphe G: a ( H ) ( l + A - ( H ) )2 (HI, pour tout sous-graphe H de G
(5.1)
(oii a ( H ) est la stabilitk maximum et A - ( H ) est le degrk intkrieur maximum) n’est pas suffisante pour assurer la noyau-perfection de G: exemples des graphes 71.2,s et 111,2,4(voir Fig. 1). 5.2. I1 existe des graphes noyau-critiques (voir Section 1.2) G tels que le graphe inverse G - ne soit pas noyau critique (et il existe des graphes fortement connexes sans noyau dont les graphes inverses ont des noyaux): le graphe dkcrit par la Fig. 2 en est un exemple. 5.3. Les graphes 71.2 et lll,2.4 sont, mis B part les circuits impairs, les seuls exemples connus de graphes antisymetriques minimaux sans noyau. En existe-t-il d’autres? La seule propriktk connue de ces graphes minimaux sans noyau est leur forte connexitk (voir [4,5]).
Note added in proof. La rkponse B la equation 5.3 est affirmative; voir [9].
Bibliogaphie [l] C. Berge, Graphes et hypergraphes, l t r e tdn., (Dunod, Paris, 1970). [ 2 ] C. Berge, Graphs and Hypergraphs, 2nd edn. (North-Holland, Amsterdam, 1976). [3] V. ChvBtal, Siminaire, Paris (1976).
Graphes noyau-parfaits
101
[4]P. Duchet, Reprksentations, noyaux en thtorie des graphes et hypergraphes, ThBse, Paris (1979). [5] P. Duchet et H. Meyniel, Une gtntralisation du thkorkme de Richardson sur I’existence de noyaux dans un graphe, Discrete Math., 2i paraitre. [6] H. Jacob, Thbse 3Bme cycle, Paris (1979). [7] M. Richardson, Solutions of reflexive relations, Ann. of Math. 58 (1953) 573-580. [8] J. Von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior. (Princeton Univ. Press, Princeton, NJ, 1944). [9] P. Duchet et H. Meyniel, A note on kernal-critical graphs, Discrete Maths, 2i paraitre (1981).
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Annals of Discrete Mathematics 9 (1980) 103-106 @ North-Holland Publishing Company.
THRESHOLD NUMBERS AND THRESHOLD COMPLETIONS by P.L. HAMMER Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont. N2L 361, Canada
T. IBARAKI Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto, Japan
U. PELED Computer Science Department, Columbia University, New York, NY 10027, U.S.A. Let F(x,, . . . , x,) be a Boolean function nondecreasing in each variable. F is said to be k-threshold when k is the least integer such that for some k x n matrix A and k-vector b, F ( x )= 0 iff Ax b. Lower bounds on k are established and used to construct cases where k = l/n(&,). F is called graphic when there exists a graph G with n vertices such that F ( x )= 0 iff x represents an independent set of vertices of G. This unique G is said to be k-threshold when F is k-threshold. Chvital and Hammer have characterized the 1-threshold graphs and have shown that finding the k of a graph G is NP-hard, and is equivalent to partitioning the edge-set of G into the least number of subsets each contained in some I-threshold subgraph of G. We generalize their result by characterizing the subsets of edges with that property. This result has been used to give sufficient conditions for a graph to be 2-threshold.
Every system Ax S b of linear inequalities in 0-1 variables x = (xl, . . . , x,) represents a Boolean function f:{0, 1.)”+{0, 1) such that f(x)=0 if and only if Ax S b. Of course, any function can be represented by many systems. In particular, f is called positive if it has a representation with a nonnegative A, and threshold if it can be represented by one linear inequality. We shall consider here the problem of representing a given positive function f by a system with a minimum number m of inequalities, or equivalently of expressing f as a Boolean sum g(’)+ . + g‘”’ of threshold functions g(*) with a minimum m. This m is called the threshold number of f . For a heuristic using a perturbation technique of the simplex method see Peled [ 5 ] . Another way to specify a positive function f is to describe its prime implicants, the minimal 0-1 points p such that f ( p ) = 1 . “Minimal” means here that q S p and f(q) = 1 imply q = p, where q s p means qi S pi for i = 1, . . . , n. If p is a prime implicant, the monomial xi is also called a prime implicant of f. It can be shown (for the proofs see Hammer et al. [2]) that if Ax C b represents a positive function, then A’xSb represents the same function, where A’ is obtained from A by replacing its negative entries by zeros. It follows that in our problem the minimum value of m does not increase if the g(*’ are constrained to be positive threshold functions. This gives the prime implicants of f a natural role.
-
np8=,
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For any subcollection S of prime implicants of f , let f s denote the Boolean sum of the prime implicants in S, and call S a threshold subcollection in f when there exist a threshold function g such that f s S g Sf.
Theorem 1. A positive Boolean function f is the Boolean sum of m threshold functions i f and only i f the collection of prime implicants o f f can be covered by m threshold subcollections in f. This reduces our problem to a combinatorial one, albeit very difficult: cover the collection of prime implicants of f by a minimum number of threshold subcollections in f . The main difficulty here, besides solving the inherent set-covering problem, is to recognize the threshold subcollections in f . Let us say that S is k-summable in f when there exist 2 k 0-1 points a'"? b") ( i = 1,. . . , k ) , with possible repetitions, such that fs(a'") = 1, f(b'")= 0 ( i = 1 , . . . , k ) and a ( i )= b'i),where the sum denotes vector addition. If S is not k-summable, it is called k-asummable in f. It is a consequence of the separation theorem for polytopes that S is a threshold subcollection in f if and only if S is k-asummable in f for all k . This is the only known characterization and it is hard to use. Therefore we shall relax the condition on S to 2-asummability in f, which is easier to deal with and will provide useful lower bounds on the optimal m. Let p'l), . . . , p'k' be the prime implicants of the positive function f . Define a graph G, on the vertex set (1,. . . , k } , in which i and j are adjacent precisely when { P ' ~ ) p"} , is 2-summable in f.
xi
xi
Theorem 2. With the notation above, a subcollection S of prime implicants o f f is 2-asummable in f if and only i f S is an independent set of vertices in Gp Consequently the chromatic number of Gf is a lower bound on the threshold number of f . In some cases this is equal to an upper bound. For example, suppose that for any two distinct prime implicants p , q of f there exist two components i, j such that pi = pi = 1, qi= qi = 0. Then it can be shown that G, is a complete graph, and consequently the threshold number of f is at least the number of prime implicants. But obviously each prime implicant is a threshold function in itself, and so the bound is met. Coding theory provides other bounds on m when in addition all the prime implicants have the same length, i.e. number of components equal to 1. In particular the threshold number can then be as large as l / n ( & , ) , but by Sperner theorem it must be less than (Ln;Z,) for any positive f . If f is not required to be positive, its threshold number can be as large as 2"-' but not larger (see Jeroslow [4]). We now specialize to the case of a positive f all of whose prime implicants have length 2. In that case one associates with f a graph G on the vertex set (1, . . . , n } whose edges correspond to the prime implicants of f, and then f ( y ) = 0 if and only
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if y is the characteristic vector of an independent set of vertices of G. Conversely, any graph arises in this way, and for this reason a positive f all of whose prime implicants have length 2 is called graphic. The following result characterizes Gf in this case.
Theorem 3. Let f be graphic. Then two prime implicants xixi and XkXl of f are adjacent in Gf i f and only if both xixk and xjxl or both xixl and XjXk are not prime implicants of f . Thus by Theorem 2 and 3, if the graphic function f is associated with the graph G, then a set S of edges of G is 2-asummable in f if and only if S does not contain the configuration
where solid lines indicate edges from S and dotted lines indicate non-edges of G. In particular, the set of all edges of G is 2-asummable in f if and only if G, has no edges. In fact, Chvatal and Hammer [l]showed that a graphic function f is a threshold function if and only if Gf has no edges. They also showed that the threshold number of a graphic f is NP-hard to compute, and that it is not less than the chromatic number of G,, which is a special case of our results above. If a graphic f is associated with G, and S is a subcollection of the edges of G. then the condition that S be 2-asummable in f is necessary but not sufficient for S to be threshold in f , as the following example demonstrates: f = x , x * + x ~ x 3 + x ~ x 4 + x 4 x 5 + x 5 x l+x2x,+x3xs
S is independent in Gf and hence 2-asummable in f, but there does not exist a aixiS b represent threshold function g satisfying fs S g Sf.For otherwise let
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g. Then since g(1, l , 0 , 0 , 0 ) 3 f s ( l , 1 , 0 , 0 , 0 ) = 1 and g(0, 1 , 0 , 0 , 1)s f (0, 1 , 0 , 0 , 1)= 0 we have a , > a,, and similarly a, > a3> a,, which is a contradiction. The main result below characterizes those subcollections of edges of G associated with a graphic f that are threshold in f.
Theorem 4. Let f be a graphic Boolean function associated with a graph G and let S be a subcollection of edges of G. Then S is threshold in f if and only if G does not have vertices vo, vl,. . . ,v2,-,, not necessarily distinct, such that for all j (vZi,vZjfl) is an edge in S and (vZj+,,vZit2) is a non-edge of G (indices are modulo 2t). The necessity of this condition can be easily proved in anology with the example above. The sufficiency can be proved by an (efficient) algorithm to find a set T of edges of G, containing S, such that fT is a threshold function. In the special case that f is graphic and G, is bipartite, a simpler equivalent condition can be given. It is obtained from the one in Theorem 4 by fixing t to be 3 and requiring all of vo, . . . , v5 to be distinct, except possibly v,, and v3. This has been exploited in [ 3 ] to give conditions under which a graphic f with bipartite Gf has threshold number 2.
References [ 13 V. Chvfital and P.L. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math. 1 (1977) 145-162. [2] P.L. Hammer, T. Ibaraki and U.N. Peled, Threshold numbers and threshold completions, Dept. of Combinatorics and Optimization, University of Waterloo, CORR 79-41 (1979). [3] T. Ibaraki and U.N. Peled, Sufficient conditions for graphs with threshold number 2. Dept. of Combinatorics and Optimization, University of Waterloo, CORR 78-13 (1978). [4] R.G. Jeroslow, On defining sets of vertices of the hypercube by linear inequalities, Discrete Math. 11 (1975) 119-124. [5] U.N. Peled, Regular Boolean functions and their polytopes, Thesis, Dept. of Combinatorics and Optimization, University of Waterloo (1975).
Annals of Discrete Mathematics 9 (1Y80) 107-119 @ North-Holland Publishing Company
QUASIMONOTONE BOOLEAN FUNCTIONS AND BISTELLAR GRAPHS* Peter L. HAMMER Department of Combinatorics and Optimization, University of Waterloo, Ont. Canada, and IRMA, Uniuersitk Scientifique et Midicale de Grenoble. France
Bruno SIMEONE Zstituto “M. Picone” per le Applicazioni del Calcolo, CNR. Rome, Italy, and Department of Combinatorics and Optimization. Uniuersity of Waterloo, Ont. Canada A bistellar graph is defined as a graph whose edge-set can be partitioned into stars so that each vertex is incident to at most two stars. A structural characterization of bistellar graphs is given: they are seen to be closely related to “almost monotone” boolean functions, as well as to injective graphs. It is shown that the maximum independent set problem for bistellar graphs is NP-complete. Implications on the computational complexity of quadratic 0-1 optimization problems are discussed.
1. Introduction The present paper deals with the class of those graphs, whose edge-set can be partitioned into stars (i.e. complete bipartite graphs of the form K1,,) in such a way that each vertex is incident to at most two stars. Such graphs will be called bistellar. The study of these graphs has been motivated by the following question in discrete optimization. Consider the problem of maximizing over the binary n-dimensional hypercube B“ (where B = (0, 1)) the quadratic function f(x) = xTQx. Without loss of generality, we may assume that the matrix Q is upper triangular. It is well-known [ 5 ] that, when all off-diagonal elements of Q are non-negative, the above problem is reducible to a maximum flow one, and hence can be solved in polynomial time. We might then ask whether the problem is still polynomially solvable under the slightly more general assumption that Q has at most one negative coefficient per row (but not necessarily on the diagonal). The conclusive result of this paper states that the answer is negative, unless P = N 9 . ’ Such result is a direct consequence of the NP-completeness of the maximum independent set problem for bistellar graphs, which will be established in Section 4.
* Presented
also at the Oberwolfach Meeting on Mathematical Programmings, May 1979 see [I].
’ For the definition of terms like ??, ”9, NP-completeness. 107
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Section 2 gives a structural characterization of bistellar graphs. Actually, they are seen to be closely related to “almost monotone” boolean functions, as well as to injective graphs (see e.g. [2]). Some inequalities involving arithmetic parameters of bistellar graphs, such as number of vertices, number of edges, etc., are also derived. In preparation to the above mentioned NP-completeness results (Section 4), some reductions for the maximum independent set problem in arbitrary graphs are given in Section 3.
2. Characterization and properties of bistellar graphs We adopt here the boolean-theoretic terminology of [6]. Consider a disjunctive boolean form in n binary variables xl,. . . , x, a
= T ,V ’
. .vT,,,,
where each term Th is a finite product of literals, i.e. variables xi or their complements = 1 -xi. Let us associate with a a graph G,, the conflict graph of a, as follows. The vertices of G, are the terms of a, and two vertices Th and Tk of G, are adjacent if and only if, as terms, they have at least one conflict uariable xi, i.e. a variable which is complemented in Th and uncomplemented in Tk,or vice-versa. Conversely, given a graph G, a (conflict) code of G is any disjunctive form a such that G is the conflict graph of a. A boolean function a is called quasimonotone if it has a disjunctive form in which each complemented variable appears at most once. A characterization of quasimonotone quadratic boolean functions is given by Proposition 1 below. We recall that a boolean function a : B“ + B is monotone (increasing) in the variable xi if, for all 0-1 values u l , . . . , v i - l , u ~ +. .~. ,, v,, one has ff(UI,.
. . , ui-,, 1, ui+*, . . . , u,)>a(u1,. . . , U i & l , O . Ui+l,. . . , v,).
Proposition 1. A quadratic boolean function a is quasimonotone i f and only if, for each variable xi, there is a variable xi such that the restriction of a to the hyperplane xi = 0 or to the hyperplane xi = 1 is monotone in xi. Proof. Is a direct consequence of the well-known fact [6] that a boolean function a is monotone in the variable xi if and only if a has a disjunctive form in which xi is never complemented (if a is quadratic the disjunctive form can be taken to be quadratic as well). If G is a graph, a partition of its edge-set into stars is called quadratic if each
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vertex of G is incident to at most two stars. A graph is bistellar if it admits a quadratic partition of its edge-set into stars. The relationship between quasimonotone boolean functions and bistellar graphs is given by
Proposition 2. A graph is bistellar i f and only i f it has a quasimonotone quadratic code.
Proof. Let the graph G have a quasimonotone quadratic code. Without loss of generality, we can assume that the code has the following properties: (a) n o variable is a dummy, i.e. each variable appears both in complemented and uncomplemented form; (b) any two conflicting terms have exactly one conflict variable. Indeed, we can always erase all dummies, so that (a) necessarily holds; of course, conflicting terms will still be conflicting. On the other hand, if two terms conflict in two variables, they are necessarily of the form xy and Xy. Since the code is quasimonotone, there cannot be any third term involving X. Hence, by replacing xy by 7, one obtains a new (quasimonotone quadratic) code in which the two new terms y and Xy satisfy (b). Since a is quasimontone, under assumption (a) and for every variable x, the subgraph of G induced by the vertices involving x or X is a star centered in the (unique) vertex where X appears. Because of (b), every edge of G, belongs to exactly one star. Finally, since the code is quadratic, at most two stars meet at each vertex. Hence G is bistellar. Conversely, let G be bistellar, and let r = { S ' , . . . , S,} be a quadratic partition of the set of edges of G into stars. Let us introduce, for each star Si, a variable xi, and, for each vertex u in the star Si,let us give to u the label Xi or xi according as u is the center of Si or not. In this way, each vertex will receive at most two labels, since G is bistellar. Hence, by associating to each vertex the product of its labels, one obtains a quadratic code, which is obviously quasimonotone (Fig. 1). x x 1 2
x x
x x
x x
3 4
3 4
1 2 Fig. 1.
Remark. If G is bistellar, we can always find a quadratic partition of its edges into stars such that every vertex u is the center of at most one star: if this is not the case u is the center of exactly two stars and we can merge such two stars into a single one. From the proof of Proposition 2 it follows that every bistellar graph has a code in which no term has two complemented variables.
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If G is a graph, let us call a vertex v of G light or heavy according as u has degree ~2 or 3 3 in G. A subgraph of G (in particular an edge, a path, and so on) will be called heavy if all its vertices are heavy in G. The heavy subgraph of G is the subgraph induced by the heavy vertices of G. Light subgraphs can be similarly defined. A graph is said to be injective if its edges can be oriented so that every vertex has indegree at most one.
Lemma 3. If G is a bistellar graph and rr is any quadratic partition of the edges of G into stars, then every heavy vertex of G is the center of a star of 7 ~ .
Proof. At least one of the endpoints of each edge must be the center of a star; thus, if the heavy vertex v were not the center of a star, v would be adjacent to at least three centers of different stars, and rr would not be quadratic.
Theorem 4. A graph is histellar if and only if its heavy subgraph is injective. Proof. Let G be bistellar, let H be its heavy subgraph and let rr be a quadratic partition of E into stars. Then, for each star, let us orient all edges in the star so that their head is the center of the star: in the orientation induced in H, no vertex u of H can have indegree d 3 2, for otherwise u would be incident to at least two stars whose centers are distinct from u, and to at least one additional star with center 0,by Lemma 3. But this contradicts the fact that G is bistellar. Conversely, assume that the edges of H are oriented so that the indegree of each vertex is at most one. Further, let us orient all edges, joining an heavy vertex u with a light vertex w, so that u becomes their head; and let us arbitrarily orient all edges having two light endpoints. For every vertex v of G, let S ( v ) be the set of all edges whose head is v. Then S ( v ) is a star and r r = { S ( v ) : V E V} is a quadratic partition of E into stars.
Theorem 5 . A graph is bistellar i f and only if every connected component of its heavy subgraph has at most one cycle. Proof. Follows immediately from Theorem 4 and from a theorem of Berge [2, Chapter 171, according to which a graph is injective if and only if each of its connected components has at most one cycle. The remainder of this section deals with inequalities involving some parameters (such as the number of vertices, the number of edges, etc.) of a bistellar graph. Let us denote by y1
-the
m -the q-the p -the
number number number number
of of of of
vertices of G, edges of G, light vertices of G, heavy vertices of G.
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Theorem 6. For any bistellar graph G, one has m S n + q. Proof. Let X be the set of heavy edges of G and 9 the set of the remaining edges of G. As a consequence of Theorem 5, each connected component of the heavy subgraph has a number of edges not greater than the number of its vertices Hence 1x1s p . On the other hand, every edge in 9 is incident in at least one light vertex and no more than two edges of 9 are incident in any light vertex. Hence 1.91 s 2q. Therefore m=IX\+19l~p+2q=n+q.
Corollary 7. For any bistellar graph, one has m 6 2 n . Corollary 8. For any bistellar graph G without isolated points or pendants, one has n63q. Proof. If d ( u ) is the degree of vertex u of G, we have d ( u ) 2 3 for all heavy vertices and d ( u ) = 2 for all light vertices. Taking into account Theorem 6, we have 3p + 2 q 6
2
d ( u )= 2 m S 2 n + 2 q = 2 p + 4 q .
O € V
Hence p s 2q, that is n 6 3q. Taking into account Theorem 5 , it is easy to see that
Proposition 9. A n y bistellar graph has Chromatic number at most 3 . In conclusion, bistellar graphs are displaying an extremely simple structure. They possess few edges, they have many vertices of degree S 2 and a low chromatic number. Thus one is led to hope that in such graphs an independent set of maximum cardinality can be determined in polynomial time. As we shall see, however, this problem turns out to be NP-complete.
3. Reductions for the maximum independent set problem In preparation to the NP-completeness result of next section, the present section includes a number of reductions for the maximum independent set problem in an arbitrary graph. Let G = (V, E ) by any graph. If S G V, the neighborhood N ( S ) of S is the set of all vertices in V- S which are adjacent to some vertex of S. An independent set is a set S of vertices no two of which are adjacent. We are interested in the problem of finding an independent set of maximum cardinality. Such maximum cardinality is denoted, as customary, by a ( G ) .
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A useful tool for establishing many of the reductions in the sequel is the following result, due to Nemhauser and Trotter [4]. Lemma 10. If S is a maximum independent set in the subgraph of G induced by S UN(S), then S is contained in some maximum independent set of G.
Proof. See [4, Theorem 13. Two vertices of G are said to be of the same kind if both are heavy or both are light. A n alternating subgraph of G is a subgraph H of G such that no two vertices of the same kind are adjacent in H.
Theorem 11. The problem of finding a maximum independent set in an arbitrary graph G is reducible (in polynomial time) to the problem of finding a maximum independent set in a graph G' such that ( I ) all light vertices of G' have degree two; ( 2 ) no two light vertices of G' are adjacent; ( 3 ) G' has no alternating path whose endpoints are adjacent. Proof. Reduction (1). If x is an isolated vertex, the reduction is trivial. If x has degree 1 and y is the (only) vertex adjacent to it, there is a maximum independent set S" of G containing x, by Lemma 10. S" is obtained by adding x to a maximum independent set of the subgraph G' induced by V - { x , y } . Reduction ( 2 ) . Let x and y be two adjacent light vertices of G. By Reduction (l), we can assume that x and y have degree two: let u be the other vertex adjacent to x and 2) t h e other vertex adjacent to y (Fig. 2). If u = u, there is a maximum independent set S" of G containing x, by Lemma 1 0 . S" is obtained by adding x to a maximum independent set of the subgraph G' induced by V {x, y, u}. If u f u, there is a maximum independent set S" of G containing at most one of t h e two vertices u, u : indeed, if u and u both are in S", by replacing u with y one obtains another maximum independent set. Hence S" is a maximum independent set also for the graph G"= G U uu.
Fig. 2.
If G' is the graph obtained from G by replacing the 3-path uxyv by the edge uu (if uu is already present, one merely deletes the vertices x and y with their stars) let S' be a maximum independent set for G'.
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At least one of the two vertices x, y, let it be x, is adjacent to no vertex in S'. Since certainly a(G")GIs'\+ 1, the set S'U x is a maximum independent set for G" and hence also for G. Thus the problem in G is reducible to the problem in G'. Reduction (3). Let P be such a path, and let d be the number of vertices on P. The light vertices along P form an independent set L of cardinality [ I d ] . On the other hand for the subgraph G ( P )induced by P, one has a ( G ( P ) ) S [ i d ] because , G(P) has a spanning cycle of length d . Therefore, L is a maximum independent set for G(P); hence, by Lemma 10, L is contained in some maximum independent set of G. It follows that the problem in G is reducible to the corresponding problem in the subgraph G' induced by the vertices of G not belonging to P. A graph G' with the properties (l), (2), (3) of last theorem will irreducible. Let s.4 be the family of all maximal2 alternating trees of a graph G For every A € & , let us denote by H ( A ) and by L ( A ) the set of vertices and the set of all light vertices of A, respectively. Further, let collection of all sets H ( A ) , A E&.
be called (Fig. 3). all heavy W be the
Fig. 3.
Lemma 12. If the graph G is irreducible, then W is a coloration of the set U of heavy vertices of G (i.e. i t is a partition of U into independent sets).
Proof. Clearly, every heavy vertex of G belongs to some H ( A ) .If two vertices of a given H ( A ) were adjacent, property ( 3 ) of irreducible graphs would be violated. Hence H ( A ) is independent for all A E ~Now . assume that H ( A ) and H ( B ) ( A # B) have some common vertex. The graph A U B is a connected alternating subgraph of G and cannot be a tree. for otherwise A and B would not be maximal. Thus A U B contains an alternating cycle and property (3) is violated. Hence H ( A ) and H ( B ) are disjoint. With respect to vertex inclusion.
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Fig. 4.
Let us delete a leaf and the (unique) vertex IJ adjacent to it in A. By Lemma 12, is light. In this way, a new tree A' is obtained, which still satisfies properties (1) and (2) (where, of course A should be replaced by A'). Let us iterate the above procedure until all light vertices have been deleted. The number of light vertices and the number of heavy ones which have been deleted are the same. The last deleted light vertex was adjacent to an heavy vertex w which was not deleted. Actually, w is the only undeleted vertex. because the set H ( A ) is independent. It follows that \ H ( A ) = \ IL(A)I+ 1 .
Given an irreducible graph G, the alternating tree graph GAof G is the graph whose vertices are the maximal alternating trees of G and where two vertices A, B E d are adjacent if some heavy vertex of A is adjacent to some (heavy) vertex of B in G. For example, the alternating tree graph of the graph G in Fig. 3 is shown in Fig. 5. 5
6
Fig. 5.
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and bistellar graphs
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Theorem 13. If G is irreducible, the problem offinding a maximum independent set of G is reducible to the problem offinding a maximum independent set of G".
Proof. First, we observe that there is a maximum independent set S" of G such that, for every A E ~the , set S" contains either all heavy vertices or all light vertices of A, but no two vertices of different kinds. Indeed, assume that S is a maximum independent set which contains both heavy and light vertices of A. Now, if 1 = \L(A ) Jone , has IH(A)I= 1 + 1 (for a proof see [ 3 , Section 4. I]). The only maximum independent set of A is the set of its heavy vertices and every other independent set has at most 1 elements. Therefore, if V(A) is the set of vertices of A, the set S ' = S f' V(A) has no more than 1 elements. Hence, by replacing S' by the set L ( A ) of the light vertices of A, which is independent and has cardinality I, one gets a new maximum independent set of G . If this procedure is repeated for all A E SQ, one eventually obtains a maximum independent set S" with the desired property. Therefore, we can restrict ourselves to those independent sets I of G with the property that, for all A E d,I fl V(A) is either the set H(A) or the set L ( A ) .If I is any such independet set, let 4 be the set of those A for which I n V(A) = H ( A ) .Then 4 is independent in G", and from the relation
lIl=
c
A€9
IH(A)l+
1 lL(A)I
A69
one obtains
Conversely, if 4 is an independent set of G", the set
I=
u H(A)) U
(As9
U L(A) (A69
is independent in G and ( I ) holds. Since the term IL(A)I in (1) is constant, I is an independent set of maximum cardinality in G if and only the associated set 4 is an independent set of maximum cardinality in G".
4. NP-completeness results We are now ready to prove the announced NP-completeness result for the maximum independent set problem in a bistellar graph.
Theorem 14. Given an arbitrary (non-singleton) connected graph K there is an irreducible bistellar graph G such that K = G". Moreover (a) the heavy subgraph of G is a path; (b) every maximal alternating tree of G is a path.
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Proof. Given K, the graph G is built in two phases. In Phase 1, a sequence of vertices of K (with repetitions) is generated. The elements of the sequence will eventually become the heavy vertices of G. In Phase 2, light vertices, as well as edges, are added. Phase 1. Step 1: Choose an arbitrary postman tour of K, i.e. a sequence of successively adjacent vertices of K ( possibly repeated) such that each edge of K is an edge of the current sequence. Step 2: For every vertex u which occurs only once in the walk, do the following. If u is the last element of the walk and w is the vertex immediately before it, add to the sequence two elements w and u. Otherwise, let w be the element immediately following u. Insert between u and w two new elements w and u : uw-+uwvw.
Step 3: If l , , 1, are the first two elements and r I , r, the last two elements in the sequence, add two new elements 1, and 1, at the beginning of the sequence and two new elements r l , r2 at the end of the sequence: 1, 1,
*
. r , r2 + I , 1, I , 1, . . . r , r, r l r,.
Phase 2. Step 4 : Connect each element of the sequence with the next one. Let 1 and r be the first and the last vertex in the final sequence, respectively. If eh is any element of the sequence such that (i) eh is neither the second occurrence of 1 nor the third last occurrence of r, and (ii) the vertex = eh occurs again later on in the sequence, then add a new light vertex s and connect s with eh and with the next occurrence ek of u. Step 5 : Add a new vertex s I and connect it with the first and with the third occurrence of 1. Add a new vertex s2 and connect it with the last and with the third last occurrence of r. End. The procedure will be illustrated at the end of the proof on a concrete example. Let us now prove that the graph G built by the procedure has the desired properties. First, we observe that all the elements of the sequence generated during Phase 1 are heavy vertices of G and that the remaining vertices of G are light. The subgraph induced by the heavy vertices of G, i.e. by the elements of the sequence, is a path. Hence G is bistellar. All light vertices of G have degree two and no two light vertices are adjacent. Hence G has properties (1) and (2) of irreducible graphs. Next, we observe that, for every alternating path P of G, all heavy vertices along P correspond to different occurrences of a same vertex 2) of K. The set of all
Quasimonotone boolean functions and bistellar graphs
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occurrences of u is independent in G: hence G satisfies also propety (3) of irreducible graphs. Note that every alternating tree of G is a path and that, for every vertex u of K , there is a maximal alternating path A such that H ( A ) is the set of all occurrences of u in G. and conversely. The basic feature of the construction is that uz) is an edge of K if and only if some occurrence of u is adjacent to some occurrence of u in G. Therefore, the alternating tree graph G" of G is isomorphic to K . It should be pointed out that the essential part of the construction is included in Steps 1 and 4; the remaining steps are minor adjustments required in order to ensure that all elements in the final sequence are indeed heavy.
Example. Consider the graph in Fig. 6. An eulerian walk is
Fig. 6.
1234142562 After Phase 1 the final sequence is 121234341425656262 and the graph G is shown in Fig. 7
Corollary 15. The problem of finding a maximunz independent set in a bistellar graph is NP-complete.
Proof. It is well-known that the problem of finding a maximum independent set in an arbitrary graph K is NP-complete [I]. We may assume that K is connected. Using the procedure described in Theorem 14, one can build in polynomial time a bistellar graph G such that K = G". If the number of vertices of K is k , the graph G has O(k') vertices. By Theorem 13, finding a maximum independent set in K is reducible to finding a maximum independent set in G. Hence the result follows. Corollary 16. The maximization ouer B" of a quadratic function xTQx, where the matrix Q is upper triangular and has at most one negative coeficient per row, is an NP-complete prohlem.
P.L. Hammer, B.Simeone
118
Proof. We shall show that the problem of finding a maximum cardinality independent set in a bistellar graph is reducible to the maximization over B" of a quadratic function xTQx, where Q is upper triangular and has at most one negative coefficient per row. Since we have established (Corollary 17) the NPcompleteness of the former problem, the thesis will follow. Indeed, let G be a bistellar graph and let Tl(x,3)v - * vT,,,(x,Z) be a quadratic quasimonotone code of G. The problem of finding a maximum cardinality independent set in G can be formulated as max [Tl(x,Z) + * XEB',
- + T,,,(x,X)].
Quasimonotone boolean functions and bistellar graphs
119
In fact, by the definition of conflict code, the optimal value of (2) is just the maximum cardinality of an independent set of G. Eliminating the complemented variables xi through the relations Xi = 1 - xi ( i = 1, . . . , n), we can re-write (2) in the form max xTQx XEB"
where the matrix Q is upper triangular. Because of the quasimonotonicity of the code, the matrix Q has at most one negative coefficient per row.
References [I] V.A. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley. Reading, MA, 1976). [2] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1972). [ 3 ] J. Edmonds, Paths, trees and flowers, Can. J. Math. 17 (1965) 449-467. [4] G.L. Nemhauser and L.E. Trotter, Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232-248. [S] J. Rhys, A selection problem of shared fixed costs and networks, Management Sci. 17 (1970) 200-207. [h] S. Rudeanu, Boolean Functions and Equations (North-Holland, Amsterdam. 1974).
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Annals of Discrete Mathematics 9 (1980) 121-123 @ North-Holland Publishing Company
MINIMAL TRIANGULATIONS OF POLYGONAL DOMAINS G.T. KLINCSEK School of Computer Science, McGill Uniuersity. X0.5 Sherbrooke Sf. W.. Montreal, Que. H3G 2K.5, Canada
1. Introduction Let V be a set of n distinct points (vertices) M , , M 2 , . . . , M, in the plane. We assume that no 3 points are collinear. This assumption is not essential (as long as not all the points are collinear); but simplifies the explanations. Let E be the family of $n(n- 1) line segments (edges) joining the vertices of V.
Definition. A triangulation T of V is a maximal subset of E in which no two edges cross each other. Clearly, in the planar graph determined by V and T each interior face is a triangle. The weight s ( T )of a triangulation T is the sum of the length of the edges in T.
Definition. The minimal weight triangulation MWT is a triangulation on V for which s ( T ) is minimal. Let M.W.T. also denote the weight of this triangulation. The problem of finding MWT presents many intriguing aspects: (a) Several very fast algorithms were proposed and later proven wrong. For example, the Delaunay triangulation (which can be obtained in nlog n time) [3] does not always give the exact answer [I]. (b) For the heuristics in use, little information is available about their error [ 2 ] . (c) Better analyzed problems, like the minimal-spanning-tree and the minimalHamiltonian circuit problems do not give helpful information. There are examples to show that the MWT needs not to contain the minimal spanning tree or any Hamiltonian circuit [11. (d) There is some evidence to suspect that the problem is NP-complete, but no proof to date is available El]. To gain some insight of the general problem this paper proposes to solve a variant of the intitial problem, as described in the next paragraph. 121
122
G.T. Klincsek
2. Restricted minimal triangulations Let S be a given subset of E where no two edges of S cross each other. Then there exists some triangulation T such that S c T.
Definition. The restricted triangulation problem consists of finding a T of minimal weight among those containing s. If S is a connected spanning graph over the vertices of V, the problem is solved using Algorithm B of Section 4. Since any triangulation contains the convex hull Co of the graph (V, E ) , we can start with So= S U C”. And hence the initial condition can be relaxed as: let S U C” be connected and spanning. As So separates the plane into a number of connected regions, we can apply Algorithm B to each simple domain and our answer is the union of the individual triangulations. Hence we have a rule of thumb to improve a given triangulation: select a set of n edges of the given triangulation. These should contain all the edges of C0 and span V. In other words, grow a “spanning” tree from the convex hull to the interior of V. A simple polynomial domain results, for which the best triangulation can be obtained in O(n’) operations. The question of the selection of the most appropriate spanning tree is still open, since this of course, would solve the general problem.
3. Algorithm A-Triangulation
of a convex polygon
Let MI,M 2 , . . . , M,, be the vertices (ordered clockwise) of a convex polygon in the plane. To emphasize this ordering we will use the name Mn+i for the node M i whenever Mi is reached “the second time around the perimeter”. The M.W.T. can be obtained using dynamic programming. Let C(i,j), where ( i < j ) , be the M.W.T. of the subgraph involving the nodes Mi, Mi+1, . . . , Mi. Intuitively speaking, we cut off an area of the polygon along the segment MiMi and compute the M.W.T. of this piece. ~
Algorithm A Step 1: For k = 1, i = 1 , 2 , . . . , n and j = i + k let C(i, j ) = d ( M i ,Mi), where d(Mi,Mi) is the length of the segment MiMi. Step 2: Let k = k + l . For i = 1 , 2, . . . , n and j = i + k let
(*I
C(i. j ) = d(M,,Mi)+min [C(i,m ) + C(m,j)]. i<m<j
For each pair ( i , j ) let I is achieved.
= L(i,j )
be the index where the minimum C(i,j ) in (*)
123
Minimal triangulations of polygonal domains
Step 3 : If k < n go to Step 2. otherwise the weight of M.W.T. is C(1, n ) . Step 4: To find the edges involved in the M.W.T. we should backtrace along the pointers L. The edge M , M , is in M.W.T. Step 5 : For each M i M j ~ M . W . T with . j > i + l let l = L ( i , j ) , then M,M,E M.W.T. and M,M,E M.W.T. ~
~
4. Triangulation of a simple polygon P domain. Algorithm B Let M I ,M 2 , . . . , M,, be the vertices of a simple polygonal domain D, the vertices are numbered sequentially along the boundary. (By simple we mean that D is simply connected.) As before we introduce the names For each segment MiMi ( j > i + 1 ) we need a decision. MiMj is interior to D if the line-segment MiMj (not the straight line through Mi and Mi) divides D in exactly 2 components. To find the M.W.T. for the interior of D we modify the distance function as follows: ~
d"(Mi,Mi) =
d(Mi, Mi)if j = i + 1 or if MiMj is interior to D, otherwise.
Algorithm B is the same as Algorithm A but substitute d" for d in Step 2.
5. Running time analysis
In Algorithm A the longest executing stage is Step 2. It requires a constant multiple of n x k operations for each k = 1 , 2 , . . . n. Hence the total running time of the algorithm is of order n3. In Algorithm B the evaluation of d" which may be done parallel to Step 2 or in a set-up stage, needs again at most O(n3)operations. For each of the n x $(n - 1) segments MiMj test for intersection with the O ( N ) edges of the polygonial domain D. If there is intersection, MiMj is not interior to D. Otherwise MiMj is interior to D provided it is interior to the angle Mi 1 MiMi+ 1 . ~
References [ I ] E.L. Lloyd. On triangulations of a set of points in the plane, Proc. lXth Annu. Symp. on Foundations of Computer Science. IEEE 228-240. [2] G.K. Manacher and A.L. Zobrist, Neither the Greedy nor the Delaunay Triangulation of a planar point set approximates the optimal triangulation, Information Processing Letters 9 (1979) 3 1-34. [3] M.I. Shamos, Geometric complexity, Proceedings of the Seventh ACM Symposium on the Theory of Computing (1975) 224-233.
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Annals of Discrete Mathematics 9 (1980) 125-126 @ North-Holland Publishing Company.
GRAPHS, GROUPS AND MANDALAS Paul C. KAINEN Bell Laboratories, Holmdel, NJ 07733, USA
Abstract
A class of geometries, called Mandalas, have been of recent interest for their applications to human visual perception. Consider two interacting opposite-sense circular forces F, and F, with unequal magnitudes IF,I >IF,\ and harmonically related angular velocities 7, and y,, respectively, with - yJyr = j/k, where 1 <j < k are integers with no common divisor. The resultant of these forces is a mandala M(j, k ) and it can be proved that M(j, k ) has j + k extreme points and j “windings”. In this paper it is shown how to construct the topology of such a mandala using a well-known graph/group method.
PACKING PROBLEMS Claude BERGE Universite‘ de Paris Vl, Pans, France
Abstract
Most of the optimization problems in integers can be reduced to a large class of combinatorial problems, often called “packing problems”, which are defined as follows: Let H = (El, E,,. . . ,Em>be a finite family of sets; let H’ = Ei2,. . . ,Ei,)be a subgraph of H whose members are pairwise disjoint (“matching”); find a matching of maximum cardinality. We shall denote by v ( H ) the maximum number of members of H which are pairwise disjoini. A subset T of UEi i s called a transversal set of H if T meets all Ei’s:find a transversal set of minimum cardinality. We shall denote by T ( H )the transversal number of H, i.e. the smallest size of a transversal set of H. Clearly, for each matching H’ and for every transversal set T, we have
(a,,
125
126
Abstracts
~T~>~H So,’ ~if . we have obtained a matching H’ with k elements and a transversal set T with cardinality k , we know that this matching H’ is a maximum matching (and that T’ is a minimum transversal). However, this criterium is not valid for every set family H. In fact a family H for which v ( H )= T ( H ) is said to have the “Konig Property”, and a large part of Hypergraph Theory is devoted to the structural properties which yields the Konig property. In this paper, we shall consider a refinement of the classical Hypergraph Theory which can be used to show that a matching of H is maximum even if H does not satisfy v = T , i.e. if H has not the Kong property.
COMBINATORIAL DECOMPOSITION AND GRAPH REALIZABILITY J. EDMONDS Uniuersity of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
W. CUNNINGHAM Carleton Uniuersity, Ottawa, Ontario, K l S 5R6, Canada
Abstract A theory of decomposition of combinatorial structures, such as graphs, matroids, and hypergraphs, is outlined. The main results are uniqeness theorems and efficients decomposition algorithms. Recent work on finding network structure in linear programs (Bixby and Cunningham) includes a new algorithm for graph realizability which uses the decomposition theory.
Annals of Discrete Mathematics 9 (1980) 127-133 @ North-Holland Publishing Company
MULTIPLIERS OF SETS IN FINITE FIELDS AND Z, Edward A. BERTRAM Department of Mathematics, University of Hawaii, Honolulu, HI 96822, U.S.A. Let R be a finite commutative ring with identity, and S a proper subset of R, with cardinality (ISl) 3 2. Definifion: A ( S ) = { a E R I S + a = S } and M ( S ) = { b c R ) b S = S + c for some C E R } .Each b E M(S) is called a (ring) mulfiplier of S , and S t c is called a translare of S. The first two propositions were inspired by related facts from the theory of difference sets. Proposition: Either (a) or (b) below is sufficient to insure that each t E M ( S ) fixes exactly as many translates of S as it does elements of R. (a) A(S) = {O} and, for some b E M ( S ) , b - 1 is IRI)= I . invertible in R. (b) (IS], Proposition: Let S be any multiplicative group mod u, such that A(S) = {O). If there exists an integer f E S such that ( t - l , u ) = 1, then M ( S ) = S . Finally, in R =GF(p"), we relate IA(S)l, IM(S)l, and IS(,using a specially created balanced incomplete block design. One result is the following, which allows the formation of some interesting sets S with restricted multipliers. Proposition: Let S be a union of rn of the distinct translates (i.e. cosets) of an additive subgraph H t GF(p"), IH( = p k . k 2 1. If (p, m ) = ( p k - 1 , m - 1)= 1, then IM(S)l I rn.
1. Introduction, basic properties, first results Let R be a commutative ring with identity 1; R+ the additive group and R" the multiplicative semigroup. S is any proper subset of R.
Definition. A(S) = { a E R 1 S + a = S},Mo(S)= { b E R 1 bS = S}. M ( S )= { b E R I bS = S + c for some c E R } , the multipliers of S. S + a is called a translate of S. In this paper we begin an exploration of the relationships between S, A(S) and M ( S ) . Our results include a generalization of a theorem of R. Balakrishnan giving a sufficient condition in order that a difference set S mod u satisfy M ( S ) = S. We also use the parameter A of a specially created block design in GF(p") to give a divisibility restriction on IM(S)l when S is a union of translates of an additive subgroup c GF(p"). Propositions 1 and 2 follow directly from the definitions:
Proposition 1. A ( S + r ) = A ( S ) , A(S) is a subgroup of R+, and S is a union of translates of A(S). Thus IA(S)llg.c.d.(lSI, IRI). Also, if r is inuertible, then A(rS) = rA(S). 127
128
E.A. Bertram
Proposition 2. M ( S + r ) = M ( S ) , 1 E M,,(S)E M ( S ) , and M , ( S ) , M ( S ) are both subsemigroups of R x . Furthermore, Mo(rS) = M J S ) and M ( rS) = M ( S ) for each invertible r E R. Proposition 3. If R is finite and 3s,,s2 E S such that s, - s2 is invertible, thin M ( S ) is a subgroup of R". Proof. Since M ( S ) is a finite subsemigroup, we need only show that M ( S ) has no zero-divisors. If b E M ( S ) , rb = 0 ( r f O), then bS = S + c j 0 = rbS = rS + re. Suppose s,, s2 E s, s1 - s2 invertible. But 0 = rsl + re = rs2+ rc, so r(s, - s2)= 0 gives the contradiction. Proposition 4. Suppose M ( S ) is a cyclic subgroup of R" generated by d, and 1- d is invertible. Then some translate of S is a union of cosets of M ( S ) in R " , possibly including (0). Hence either IM(S)l I IS1 or (M(S)II IS[- 1 when S E GF(p") = R . Proof. If dS = S + r, let r = c(1- d ) . Then d ( S + c) = d S + dc = d S + ( c - r ) = S + r + ( c - r ) = S + c (that is, the multiplier d "fixes" the translate S + c ) . Thus for each t E S + c , t M ( S ) z S + c , and S + c is a union of cosets of M ( S ) , including the coset (0) if -cES.
Proposition 5. If R is a finite field GF(p") and S E R , then M ( S ) E M , , ( A ( S ) ) . Furthermore
Proof. For the first part, let a E A ( S ) and d E M ( S ) . If d # 1 , then, as in the proof of Proposition 4, there exists a translate S' of S such that dS'= S' ( S ' = S if d E M , ( S ) ) . Since A ( S ' ) = A ( S ) ,we have S ' + a = S ' , so d ( S ' + a ) = d S ' = S ' . Also, d ( S ' + a ) = d S ' + d a = S ' + d a . Thus S ' + d a = S ' and d a E A ( S ' ) = A ( S ) , so d E M , ( A ( S ) ) . For the second part, we may assume that M ( A ( S ) ) # { l } . Since A ( S ) # { O ) , M ( A ( S ) ) is a multiplicative subgroup of GF(p"), by Proposition 3. Since M ( S )c M , ( A ( S ) ) E M ( A ( S ) ) ,and M ( S ) and M , ( A ( S ) ) are also subgroups, 1M(S)II IM,(A(S))I I ( M ( A ( S ) ) I .Finally, IA(S)lI p" and IM(A(S))II p" - 1 . Since l M ( A ( S ) )#[ 1 and each multiplicative subgroup of GF(p") is cyclic, we conclude, using Proposition 4 and g.c.d. (IA(S)l, M ( A ( S ) ) I )= 1, that IM(A(S))l 1 IA(S)l- 1.
In 1971, Balakrishnan [l] gave a sufficient condition in order that any S, which is simultaneously a multiplicative group mod v and a difference set, satisfy M ( S )= S. We generalize that result with the following proposition: Proposition 6. Let S be a multiplicative subgroup of R, such that A ( S )= (0). and
Multipliers of sets in finite fields and Z,
129
suppose there exists an element d E S such that d - 1 is invertible. Then M , ( S ) = S M(S).
=
Proof. Since S is closed under multiplication and 1 E S , S G M , ( S ) G S. Thus, S = M , ( S ) . Suppose b E M ( S ) , b$ M o ( S ) .Then bS = S + a with a # 0. Consider the set a s . Since d E S and d - 1 is invertible, ad E aS and ad # a. Since d E M , ( S ) , d ( b S )= d(S + a ) = S + da, and bS = b ( d S )= S + da. Thus S + a = S + da, i.e. 0 # d a - a E A(S), in contradiction to A(S)= (0). Comment. The method used above actually shows that VS G R if M , ( S ) 5 M ( S ) , then IA(S)l3(aM,(S)I for some a d A(S). It is not necessarily true that JaM,(S)(3 2, however, when there does not exist d E M , ( S ) , d - 1 invertible. For example, let S = {1,4,7, 13) mod 15, a multiplicative subgroup. If bS = S + a, then either a = 10, a = 5, or a = 0, as is easily checked. Also A(S) = {0}, lOM,(S) = 10s ={lo}, and 5M,(S) = (5). The following examples show that it is not possible to weaken, in Proposition 6, either the assumption that S is a multiplicative subgroup of Z, or the assumption that there exists an element s E S with (s- 1, u ) = 1, and still conclude that M ( S )= S. Whether or not the,re exist u and a multiplicative subgroup S c H, such that: A(S)#{O} (so S is a union of arithmetic progressions), for some S E S (s- 1, u ) = 1, and M ( S ) # S, is still unresolved.
Examples. (i) If S ={l,4 , 6 , 9 } mod 15, then A(S) = {O}, S is a multiplicative subsemigroup of H I 5 , and s = 9 satisfies (s- 1, 1 5 ) = 1. Yet ( - 1 ) S = S + 5 , so M(S). (ii) If S = {1,4,7, 13) mod 15, then A(S)={O}, S is a multiplicative subgroup of ZI5, and each s E S satisfies (s - 1, 15)> 1. Since M ( S ) is all +(15) = 8 invertible elements of Z,5r S S M ( S ) .
ss
A basic theorem about a difference set D in an abelian group G is that each multiplier of D (generally, any automorphism of G which also permutes the translates of D ) fixes as many translates of D as it does elements of G (see e.g. [2, p. 1401). In our next proposition we record a closely related result concerning a (not necessarily difference) set S in a ring R. Recall that every known difference set D mod u satisfies g.c.d. (101,u ) = 1, and that every difference set D in R' must obviously satisfy A ( D )= (0).
Proposition 7. Either (a) or (b) below is sufficient to insure that each member of M ( S ) fixes as many translates of S as it does elements of R : (a) g.c.d. (ISl, IRI) = 1; (b) A(S) = {0} and, for some b in M ( S ) , b - 1 is invertible.
E.A. Bertram
130
Proof. We first prove that either (a) or (b) implies the existence of some translate of S, fixed by each member of M(S). In case (a), an argument due to Hall is enough (see [2, p. 1401). In case (b) we show that b fixes a unique translate S’ of S. From this it follows easily that S‘ is fixed by each t in M ( S ) ; for then b(tS‘)= t(bS’)= tS’, and tS’ is also a translate of S. Hence tS’ = S’. To show that b fixes a unique translate of S, suppose bS = S + c, and d ( b - 1)= 1. Then b ( S - c d ) = b S - bcd
= S + ( C- bed)= S
+~
( 1 b- d ) = S - cd,
i.e. b fixes S - c d . If b fixes both S + f and S + f ’ , then b ( S + f + f ’ ) = S + ( f + b f ’ ) and b ( S + f ’ + f ) = S + ( f ‘ + b f ) .Hence ( f + b f ’ ) - ( f ’ + b f ) ~ A ( S )By . assumption, A(S) = {0},so b ( f ’ - f ) = f’- f. Since b - 1 is invertible, f ’ = f, so b fixes the unique translate S‘ = S - cd. Finally, we show that whenever A ( S )= (0) and some (not necessarily unique) translate S + a is fixed by each b in M ( S ) , then each b in M ( S ) fixes as many translates of S as it does elements of R. This will finish the proof of Proposition 7, since both of these conditions are true in either (a) or (b). Suppose b S = S + c . Since b fixes S + a, bS + ba = S + a, so S + ( c + ba) = S + a. Then A ( S ) = (0) gives c = (1 - b)a. Likewise, if b(S + e ) = S + e , then c = (1- b)e. Thus b(e - a ) = e - a, or e E a + ( x 1 bx = x } . On the other hand, if e E a + { x I bx = x } , then b fixes S + e follows from
b ( S + e ) = b ( S + a + ( e - a))= b ( S + a ) + b(e - a ) = S + e . We have shown that b fixes exactly as many translates of S as there are elements in the set a + { x 1 bx = x } , i.e. I(x I bx = x } l .
Proposition 8. Suppose R is finite and S is any translate of a non-trivial subgroup H G R + . If every non-zero element of H is inuertible, then M ( S )U ( 0 ) is a subfield of R . Proof. Since M ( H + a ) = M ( H ) for each a E R, we assume that S = H . By proposition 3 , M ( H ) is a subgroup of R ” , so we need only show that M ( H ) U { O }is a subgroup of R + . So suppose that d H = H + c. Since 0 E dH, we have - c E H, c E H and H + c = H ; thus d E M ( H ) implies d H = H . If d l and d , e M ( H ) , then ( d ,+ d , ) H c H + H = H. If d , + d , # 0, then [ ( d ,+ d,)HI = [HI, since every nonzero element of H is invertible. Thus d , + d , E M ( H )U (0). Furthermore, d E M ( H ) implies dH = H, - dH = - H = H, so - d E M ( H ) , and we are finished. Proposition 9. Suppose S c GF(p”).Then S = aK + b, where K is a proper subfield and a#O, i f and only i f A ( S ) # { O }and IM(S)I=ISI-l. Proof. If S = a K + b, K a subfield, then M ( S ) = M ( S - b ) = M ( C ’ ( S - b ) ) = M ( K ) =K\{O}.
Multipliers of sets in finite fields and 72,
131
Thus IM(S)J=ISI-l.Also,
A ( S )= A ( S - b ) = A ( a K )= a A ( K )# (0). On the other hand, if A ( S )# (0) and IM(S)(= ISI- 1, then Proposition 5 yields ( S J -1 divides [ A ( S ) J1. - Since IA(S)l divides IS\, we now have IA(S)l= IS1 and S = A ( S )+ s. Thus M ( S )= M ( A ( S )+ s) = M ( A ( S ) ) . By Proposition 8, M(A(S))U{O} is a subfield, say K. Thus M(S)U{O}=K. By Proposition 4, S + b = a ( M ( S )U{O}), since IM(S)l= IS1 - 1, i.e. S + b = aK, for some a.
Example. If S c GF(p"), IS1 = pk and g.c.d. ( k , n ) = 1 , then IM(S)l divides p" - 1 and g.c.d. (ISl, p" - 1)= 1 yield IM(S)l divides IS1 - 1. Thus lM(S)I divides g.c.d. (pk- 1 , p" - l), i.e. IM(S)l divides p - 1. In particular, if S is a translate of an additive subgroup H G GF(p"), \HI = pk, g.c.d. (k, n ) = 1, then M ( S )U{O} is a subfield so M ( S )U (0) is the prime subfield of order p.
2. Block designs in finite fields If S s F = G F ( p " ) we will refine Proposition 4. Using the parameter A of a specially constructed balanced incomplete block design, we show that if lM(s)l#lsl- 1 , then
Another corollary yields a divisibility restriction on IM(S)l when S is a union of a certain number of translates of an additive subgroup of GF(p"). A balanced incomplete block design is any arrangement of u distinct objects into b (not necessarily distinct) blocks, each containing k distinct objects, such that each object occurs in exactly r of the blocks and each (unordered) pair of distinct objects occurs in exactly A blocks. The parameters of the block design may be listed as [u, k , A], since b and r are determined from these by the relations u r = b k and A ( u - l ) = r ( k - 1 ) . For U, VsGF(p"), O$U, define 9 ( S ; U,V)= { b ( S + a ) : b E U, a E V}, a collection of subsets we call blocks. We distinguish between blocks arising from different pairs (b, a ) even when the blocks are / i ( S )= identical subsets. Here 2 d (SJd p" - 1, and F" = GF(p") - { O } . {al, a * , .. . , a,} is a complete set of coset representatives for A ( S ) in F+, and G(S)= {bl, b2, . . . , b,} is a complete set of coset representatives for the subgroup M ( S ) of F".
Theorem. 9 ( S ; G(S),A(S)) is a block design with distinct blocks, and parameters
132
E.A. Bertram
Proof. We first show that W ( S ; F " , F ) is a block design (possibly with repeated blocks), with parameters [p", ISI, ISl(lS( - l)]. First, if p f 2 and P is any subset of F" such that P and - P partition F", then 9 ( S ; P,F ) and 9 ( S ; -P, F ) are each block designs (possibly identical), with parameters [p", IS1, ilSI(lS1- l)]. For let us assume that ( p f 2 and) the block b ( S + a ) of 9 ( S ; P, F ) contains the pair (0, 1). If x # y and {x, y} # (0,l}, then one of (x - y ) b or (y - x ) b is in P. Suppose (x - y ) b E P. Then (x - y)b(S + a + y/(x - y)b) is a block of 9 ( S ; P, F ) which contains {x, y}. If ( y - x ) b ~ P , interchange the roles of x and y, again obtaining a block of 9 ( S ; P, F ) containing {x, y}. On the other hand suppose c ( S + d ) E 9 ( S ; P, F ) , and contains (x, y). If c/(x - y) E P, then ( S + d-:)
E
9 ( S ; P, F )
X-Y
and contains (0, 1); if c / ( y - x ) ~ P, again interchange the roles of x and y. Thus, every pair occurs the same number of times as the pair (0, 1). Furthermore (e.g. by using ( y - x ) b instead of I x - y)b, or vice versa), we have a 1-1 correspondence between the occurrences of an arbitrary pair among the blocks of S ( S ; P, F ) and the occurrences of any pair among the blocks of Q(S; -P, F ) . Finally, suppose that the blocks of B(S;P,F ) are arranged in rows so that for each b E P one row contains the blocks b ( S + a ) , UEF.Then each element of F appears exactly IS1 times in each row. Thus 9 ( S ; P, F ) and 9 ( S ; - P , F ) are each block designs with parameters [p", (SI,$1Sl(lSl- l)], and 9 ( S , F", F ) = Q(S; P, F ) U S ( S ; -P, F ) is a block design with parameters [p", 19 .1, ISl(lSl- l)]. If p = 2 , let b ( S a ) be a block of 9(S;F", F ) containing (0, 1). Then b(S + a + b - ' ) is a distinct block containing (0, 1). So the occurrences of (0, 1) can be paired off. If (0, 1) occurs in b ( S + a ) and {x, y}#{O, l}, then (x, y) ( x f y) appears in (x+y)b(S+a+x/(x+y)b), and again in ( x + y ) b ( S + a + y / ( x + y ) b ) . We again obtain a 1-1 correspondence between the pairs of occurrences of any two pairs. As before, the number of appearances of each element of F in any row is JSI,and 9(S;F", F ) is a block design with parameters [p", ISI, ISl(lSl- l)]. To show that 9 ( S ; fi(S),A(S)) is a block design, let d E M ( S ) and a E A(S). Then 9 ( S ; d G ( S ) ,a +A(S)) has precisely the same blocks as 9 ( S ; fi(S),A(S)). To see this, let b ( S + 6 ) ~ 9 ( S ; d f i ( S ) , a + A ( S ) )Since . b = d b , , b z E M ( S ) , and ii = a + a,, a, E A(S),
+
6(S+ 6 ) = db, (S + a + a,)= db,(S + a,) = b,(dS + da,) = b, (S + c + da,)
+
= bi(S a[)
for some a, E A(S), and b , ( S + a[)is a block of 9 ( S ; fi(S),A(S)).Likewise, every block of 9 ( S ; fi(S),A(S)) is a block of 9 ( S ; dfi(S),a + A(S)). Moreover, it follows easily from the definitions of M ( S ) , fi(S),A(S), A(S) that for fixed (d, a ) the blocks within B ( S ; dfi(S), a + x ( S ) ) are distinct subsets of F. By taking all ordered pairs (d, a ) in M ( S )x A(S) we obtain a partition of the blocks of 9 ( S ; F", F ) into disjoint arrays of blocks, each array identical to
Multipliers of sets in finite fields and E,
133
a(S),
B ( S ; fi(S),A(S)). Thus the collection B ( S ; A(S)) is a block design whose parameters are obtained from those of B ( S ; F", F ) by dividing h (as well as b and r ) by lM(S)IlA(S)l.
Corollary. If S G GF(p") and (M(S)/tIS1-1, then
Proof. From the previous theorem, ISl(lSl- 1) IA(S)l lM(S)I
is an integer. By Proposition 4, since we may assume IM(S)l# 1, IM(S)l I IS1 and thus [M(S)I is relatively prime to ISI- 1. Hence IM(S)l must divide the integer PI/IACS,l.
Corollary. Let S be the union of m of the distinct franslates of an additive subgroup H c GF(p"), [HI = p k , k 3 1. If (p, m ) = ( p k - 1, m - 1)= 1, then (M(S)lI m. Proof. Since S E S + S E H + U ,h E H + h + s E ( h + H ) + a = H + a . Thus H G A(S). Since (A(S)(1 IS( and (p, (S(/(H() = (p, rn) = 1, H = A(S). W e may assume ( M ( S ) l #1. If IM(S)l 1 (SI- 1, then IM(S)l I mpk - 1. By Proposition 5, IM(S)l/JA(S)J - 1. Since (m - l ) p k = ( m p k - 1)- ( p k - 1) we obtain 1M(S)(1 ( m - l ) p k . But ( m - 1, p k - 1)= 1, so IM(S)l and rn - 1 are relatively prime. Also (JM(S)J,p k ) = 1, so JM(S)l=1, a contradiction. Thus \M(S)))C)S)-1 and the above corollary yields (M(S)lI (Sl/lA(S)(,i s . (M(S)lI m. Example. Let H be an additive subgroup of GF(p"), p an odd prime, and S = H U H + a, a $ H . From the previous corollary (m = 2), IM(S)\12. Since - l(HUH+ a ) = -HU
- H - a = HUH- a =(HUH+a ) - a,
we have - 1 E M ( S ) . Thus M ( S ) = (1,
-
l}.
References [ I ] R. Balakrishnan, Multiplier groups of difference sets, J. Combin. Theory 10 (1971) 133-139. [2] M . Hall Jr., Cornbinatorial Theory (Blaisdell Publishing Co., Waltham, MA, 1967).
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Annals of Discrete Mathematics 9 (1980) 135 @ North-Holland Publishing Company.
ANALOGUES FOR SPERNER AND ERDOSKO-RADO THEOREMS FOR SUBSPACES OF LINEAR SPACES Gil KALAI Hebrew Uniuersity, Jerusalem, Isreal
Abstract
Theorem A. Let (Vl, Wl), (V2, W J ,. . . ,(V,,,, W,) be m pairs of subspaces of n-dimensional vector F" such that: (a) Vi n Wi = ( 0 ) for 1 s i s m ; (b) Vinw.#{O}for l s i f j s m . Then m < (,$). If, moreover, dim ViS k S n/2 for i = 1,2, . . . , rn, then rn S (L).
Theorem B. Let (Vl, Wl), . . . , (V,, W,) be m-pairs of subspaces of F",satisfying (a), (b) of the previous theorem and: (c) dim Vi = k s 4 2 , (d) d i m ( V i n y ) s l for l S i # j = S m , then m s (:It).
135
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Annals of Discrete Mathematics 9 (1980) 137-139 @ North-Holland Publishing Company.
GENERALIZED PRINCIPLE OF INCLUSION AND EXCLUSION AND ITS APPLICATIONS WE1 Wandi Mathematics, Sichuan University, Sichuan, People's Republic of China
As is well known, the principle of inclusion and exclusion is very important and useful for enumeration problems in combinatorial theory. The principle deals with the following problem. Let P = {PI,Pz, . . . , P,} be a set of rn properties, A be an arbitrary set, and r be a fixed integer. The problem is to find the number of elements of A that satisfy exactly r properties of P, given the numbers of elements of A that satisfy at least k ( k 3 r ) properties of P (see [l]). In this paper we propose and solve the following problem. If the properties in the above problem are divided into n groups, and rl, r 2 , . . . . r,, are n integers, now our problem is to enumerate the number of elements of A that satisfy exactly ri properties of the ith group of properties (1 G i S n ) . We shall get the weighted formula and use it to solve the generalized mCnage problem. Let us first introduce some notation. Let Pi (1 s j s mi ; 1 S i G n ) be collections of subsets of a set A. Let every a E A have a weight w ( a ) (see [2]). Let W ( r 1 ,... , r,,) be the sum of the weights of the elements of A that belong to exactly ri subsets Pj ( l s j S r n i ) for every i ( l s i s n ) , wk,, . _ . ,k , the sum over all choices of subsets Ki of {1 ,2 ,. . . , mi} of cardinality ki (1 S i S n ) , of the weights of the elements of A that belong to at least the subsets Pi, j E Ki,1G i G n. Then we have:
Theorem 1 (The generalized principle of inclusion and exclusion).
Proof. Since it is evident that
..
.
.
(lsirn)
the theorem follows from standard techniques of generating functions or Mobius inversion. Now let us turn to its applications. J.H. van Lint [4] has generalized the mCnage problem and solved the following problem concerning the permutations with restricted position. Let Urn,,,be the 137
W.Wei
138
number of those permutations a1a2. . . a,,,+,of the set [l,m + n] such that there are no equal numbers in every column of the array 1 2 * * * m-1 m 1 m-2 a, a2 . . . %-I He has proved
m + l m+2 ... m + n - 1 m+n m + l * . . m+n-2 %+I am+2 . . . a m +n - 1
m m-1
+
= Urn+, Urn-,,
m+n m+n-1 a m +n
(m a n 3 2)
by de Bruijn’s method, where U, is the mCnage number. So enumerated through the mtnage numbers. The general form of this problem is as follows. Let m = m, * * m ia 2 (I S i S n). We denote by
can be
+ + m, and let
urnl.., m , ( r 1 , . .
>
*
(2)
rn)
the number of those permutations ala2 . . a,,, of the set [l, m ] for which there are exactly r, columns of the ith column group, such that there are equal numbers in every column of these ri columns, in the array
1 1 ... rn, 1 a , a2 * * * L *
m,
m,+l
m,-1 am,
am,+,
m,+m,
m,+2 m,+l am,+:!
1st column group
. . . m,+ . . . +m,-,+l m . . . a m l + . . . + m ,n+ I
\
...
m1+
...
am,+m,
.
...
m2
ml+m2-l
... I
2nd column group
ml+ ... +mn-,+2 ml+ . . . +mn+,+1 a m , + . . +m,,-,+2
”
... am
m m-1 I
n th column group Now our problem is to find the enumeration formula for (2). It is obvious that the van Lint’s problem is the very particular example of our problems for n = 2 and rl = r2 = 0. The de Bruijn’s method was rather complicated for solving the van Lint’s problem. It seems very difficult or impossible to apply de Bruijn’s method to the general problem (2). But we may easily apply Theorem 1 to finding the enumeration formula for (2). By the method used to prove (2.1.4) in [3], we have
Lemma. The sum over all choices of subsets Kiof { m , + + m i _ , + 1, . .. , m , + . . . + mi} of cardinality k, (1d i < n), of the number of permutations a, . . . a,,, of [ l , m ] such that in the ith column group of (3), at least the columns of Ki have a repeated number, is
Generalized principle of inclusion and exclusion
139
Let A be the set of all permutations of [I, m ] , and define the subsets Pj of A ( l s j = z m l ; l s i s n ) as follows: P;: the permutations a, a, of [1, m ] such that aml+
+m,
m , + * * * + ml-, + j , + I - -[or m,+ * . . +m,-,+j-l
,
if 2 < j S m , ,
the permutations a, . . . a,,, of [l,rn] such that am,+
-I
+ m , _ , + ~-
m,t * . * + m , - , + l 0'
m , + . . . + ml-l+m,.
Taking w ( a ) = 1 for all a E A, we have
urn,
.,,(ri,.
. . , r n ) = W r l , . . . , r,,)
and hence by Theorem 1, we have
Besides, Theorem 1 may be applied to some other combinatorial problems, such as the generalized problem of derangements. Finally, the author is greatly indebted to Professor Chao KO and the referees of this paper for many valuable suggestions.
References [l] [2] [3] [4]
J. Riordan, An Introduction to Combinatorial Analysis (Wiley, New York, 1958). H.J. Ryser, Combinatorial Mathematics, Carcus Math. Monograph 14 (1963). M. Hall Jr., Combinatorial Theory (Blaisdell, New York, 1967). J.H. van Lint, Combinatorial Theory Seminar Eindoven University of Technology, Lecture Notes in Math. 382.
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Annals of Discrete Mathematics 9 (1980) 141-145 @ North-Holland Publishing Company
LES G-SYSTEMES TRIPLES P.M. DUCROCQ et F. STERBOUL Uniuersifi de Lille I , B.P. 36, 59650 Villeneuve d’Ascq, France
0. Introduction Un Systkme Triple de Steiner d’ordre u peut 6tre caractCrisC par la propriCtC suivante: pour un point quelconque x considCrons les i ( u - 1) blocs qui le contiennent. En Climinant x de ces blocs on obtient un ensemble de paires qui sont les ar&tesd’un couplage parfait des points restants du S.T.S. Cette propriCtC suggkre une gCnCralisation: Un G-Systkme Triple (G-ST) sera tel que, pour tout x, le graphe G, dtfini comme ci-dessus est isomorphe a un graphe donni G. L’ttude de ces G-ST peut en outre se justifier par les trois points suivants: (a) Les G-Systbmes Triples oh G est un arbre permettent d’obtenir la solution d’un prob1,bme de coloration dans la thCorie des hypergraphes [6]. (b) Les G-Systbmes Triples OG G est un cycle sont des cas particuliers de (v,3,2)-BIBD sans blocs rCpCtCs. Les mCthodes prCsentCes permettent la construction de familles infinies de tels BIBD dont certains semblent nouveaux. (c) Toujours dans le cas oh G est un cycle, le problbme de I’existence de tels systkmes triples est rCsolu grice au ThCorbme de Ringel et Youngs (Conjecture de Heawood). En retour nous esperons pouvoir contribuer, grice a certaines de nos mCthodes de construction, a une dimonstration “courte” de ce thCorkme.
1. Generalites Definition 1.1. Soit H un Systkme Triple= ( X , t), oh X est I’ensemble des sommets et 5 l’ensemble des arCtes. Toute arCte est de cardinal 3. H est un hypergraphe rCgulier 3-uniforme [2]. Pour un sommet a quelconque de X appelons G, le graphe dont les sommets sont X-{a} et les arCtes sont {E-{u}/EE~,uEE}. Si, pour tout sommet a, G, est isomorphe a un graphe don& G nous dirons que H est un G-Sysdme Triple. Si en outre X est un groupe abClien J et si les translations dans J sont des automorphismes de H, alors nous dirons que H est un (G, J)-Syst;me Triple. Si J = Z , le (G, 2,)-ST est dit cyclique. 141
142
P . M . Ducrocq, F. Sterboul
Proposition 1.2. Une condition ne‘cessaire d’existence d’un G - S T d’ordre n est que: 3 m ( H )= n * m ( G )
ou m ( H ) (resp. m ( G ) )est le nombre d’arztes de H (resp. de G ) , et n le cardinal de X.
I1 s’en suit immidiatement la condition nCcessaire d’existence: n . m ( G )= 0 (mod 3 ) . D’autres conditions d’existence peuvent 6tre donnCes comme des conditions suffisantes de non-existence. En particulier:
Proposition 1.3. Si G satisfait aux conditions suiuantes: (i) I1 existe un sommet x de degre‘ unique. Appelons p,, p 2 , . . . , ph le nombre de sommets de degre‘ d l , d 2 , . . . , d , adjacents a x. (ii) I1 existe K c { 1 , 2 , 3 , . . . , h } tel que: -1iEKPi >CidI
G tels que d ( y ) = di et d ( t )= d j OM i et j
E
K,
alors Alors il n’existe pas de G - S T .
Une caractkrisation des (G, J)-ST peut 6tre donnCe:
Proposition 1.4. H est un (G,J)-ST si et seulement si il ue‘rifie la proprie‘te‘ suiuante: ( i , j ) E G 0 - ( - - i , j - i ) ef ( i - j , - j ) € G 0 (OM
0 est le ze‘ro de J ) .
Le cas ou G est un arbre a Ctt CtudiC dans [6] et il est montri:
Proposition 1.5. Une condition necessaire et suffisante d’existence d’un arbre G tel qu’il existe un ( G , J)-ST est que n = 0 ou 2 (mod 3 ) . 2. Cycles-Systemes Triples (C,,-,-ST)
Dans ce cas m( G )= n - 1. On a donc:
Proposition 2.1. Une condition ne‘cessaire d’existence d’un C,- -ST est n = 0 ou 1 (mod 3 ) . Nous verrons au Paragraphe 7 que cette condition est aussi suffisante. En utilisant une mCthode proposCe par Doyen [ 3 ] nous avons pu dkterminer une famille infinie de C6r+2-ST, pour t 3 1.
Les G-Systtmes Triples
143
3. Chaines-Systemes Triples (P,-,-ST) Ici r n ( G ) = n - 2 et on a:
Proposition 3.1. Une condition ne‘cessaire d’existence d’un P,-,-ST est: n = 0 ou 2
(mod 3).
4. Relations entre cycles et chaines Dans certains cas il est possible de dkterminer des mCthodes de construction de Cycles-ST h partir de Chaines-ST et rCciproquement. Par exemple, sous certaines conditions, on pourra en ajoutant des arZtes a un P,-,-ST obtenir un C,-,-ST, en ajoutant un sommet et des arZtes obtenir un C,,-ST. Les constructions inverses sont Cgalement possibles. De mZme des mkthodes de construction par induction permettent de montrer:
Proposition 4.1. Si il existe unP,,-,-ST, il existe un C2,-,-S7. Proposition 4.2. Si il existe un P , _ , - S 7 et que l’une ou l’autre des conditions suivantes est ve‘rifie‘e: (i) n est pair, (ii) n est impair et pur tout sommet a, si x et y sont les extrimitis de la chaine G,. alors a et y (resp. x ) sont les extrimitisde G, (resp. G,,), alors il existe un P,_,-ST OU k = 3A . n pour tout A > 1. L’application rCpCtCe de ces constructions permet d’obtenir de nouvelles familles infinies. En particulier si I’existence de C,_,-ST est prouvke pour n = 1 ou 13 (mod 18), il est possible de dCmontrer rkcursivement a partir des rksultats prCc6dents que les conditions nCcessaires d’existence des €‘,-,-ST et des C,-,-ST sont aussi suffisantes.
5. Roues-Systemks Triples (W,-,-ST) On a alors r n ( G ) = 2 . ( n - 2 ) . D’oh:
Proposition 5.1. Une condition ne‘cessaire d’existence d’un W,- ,-ST est n = 0 ou 2
(mod 3).
En utilisant une construction proposCe par Wilson [8], nous avons pu montrer:
144
P . M . Ducrocq, F. Sterboul
Proposition 5.2. 11 existe des W,-,-ST pour tout n = m + 1 tel que: (i) m est premier, m > 3. (ii) m = 3 (mod 4), (iii) -2 est une racine primitive de Z”,.
6. Cas des graphes orient& La dCfinition des G-ST peut s’etendre a des Hypergraphes orient& fi. On obtient alors des G-ST, oh G est un graphe orientC. Nous Ctudions actuellement comment les rCsultats prCcCdents sont transformks dans ce cas.
7. Correspondance avec le theoreme de Ringel et Youngs Les travaux de Ringel et Youngs [4], concernant la dimonstration de la conjecture de Heawood, font intervenir certains systkmes de triples, mais de faGon implicite et partiellement cach6e par la thCorie des “graphes de courants”. Plus rtcemment Alpert [l], White [7] ont CtudiC B nouveau la relation entre surfaces et configurations. Alpert dkmontre:
Theoreme 7.1. Les ( n ,3,2)-configurations sont en correspondance bijective auec les repr6sentations triangulaires du graphe complef K , sur les pseudosurfaces gdniralise‘es. Appelons S ( H ) la pseudo surface en correspondance avec une ( n , 3 , 2 ) configuration H . On peut montrer:
Theoreme 7.2. Une condition ne‘cessaire et sufisante pour que S ( H ) soit une surface est que H soit un C,-,-ST. On peut donc dCduire de ce qui prickde:
Theoreme 7.3. Pour qu’il existe un C,-,-ST il faut et il sufit qu’il existe une reprisentation triangulaire de K , sur une surface S.
En outre la dCfinition classique de I’orientation des surfaces entraine: Theoreme 7.4. L a surface S est orientable si et seulement si le C,_,-ST associ6 peut &re orient6 en un (circuit)-ST. Le problkme de l’existence des C,_,-ST et des (Circuits)-ST est alors r6solu g r k e au ThCorkme de Ringel et Youngs (Conjecture de Heawood):
Les G-Systkmes Triples
145
Theoreme 7.5. 11 existe une repre‘sentation triangulaire de K,, sur une surface orientable si et seulement si n = 0, 3 , 4 , 7 (mod 12). I1 existe une reprisentation triangulaire de K,, sur une surface non orientable si et seulement si n = 0, 1 (mod 3). Compte tenu des rCsultats prCcCdents on a donc:
Theoreme 7.6. ll existe un C,-,-ST si et seulement si n = 0, 1 (mod 3). 11 existe un enp1-ST si et seulement si n = 0 , 3 , 4 , 7 (mod 12). Le thCor6me de Ringel et Youngs permet donc de rCsoudre la question de I’existence des C,-,-ST mais le probl6me n’est pas clos pour autant. En effet plusieurs de nos mCthodes de construction, bien que trks voisines de celles de Ringel et Youngs, semblent plus gCnCrales et nous espCrons pouvoir donner dans un article a venir une dCmonstration “courte” de la conjecture de Heawood.
Bibliogaphie [ I ] S.R. Alpert, Twofold Triple Systems and graph imbeddings, J . Combin. Theory 18 (A) (1975) 101 - 107. [2] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970). [3] J. Doyen, Constructions of disjoint Steiner Triple Systems, Proc. Am. Math. SOC.32 (1972) 409-416. [4] G. Ringel, Map Color Theorem (Springer-Verlag, Berlin, 1974). [5] F. Sterboul, A problem on Triples, Discrete Math. 17 (1977) 191-198. [6] F. Sterboul, A problem in constructive combinatoric and related questions, Colloq. Math. SOC.J. Bolyai 18, Combinatorics (North-Holland, Amsterdam, 1978). [7] A.T. White, Block designs and graph imbeddings, J. Combin. Theory 25 (B) (1978) 166-183. [8] R.M. Wilson, Some partitions of all triples into Steiner Triple Systems, dans: Hypergraph Seminar, Lecture Notes in Mathematics 41 1 (Springer-Verlag, Berlin, 1974) 267-277.
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Annals of Discrete Mathematics 9 (1980) 147-154 @ North-Holland Publishing Company
COMMENT CONSTRUIRE U N GRAPHE PERT MINIMAL F. STERBOUL et D. WERTHEIMER Uniuersiti de Lille 1 , 50650 Villeneuue d’ascq, France Plusieurs auteurs se sont dkja intkresses a ce problbme. Notre contribution est la suivante: -nous ne considerons que des problkmes d’ordonnancement, -nous reprenons certaines notions dCji dCfinies, ou nous en definissons d’autres qui sont voisines. De nouvelles propriCtCs (notamment le ThCor6me I ) nous permettent de proposer un algorithme de construction d’une graphe PERT minimal quant ?I la cardinalitt des sommets. This problem has already been investigated by several research workers. Our contribution may be summarized as follows: -we shall restrict ourselves to scheduling problems, -we shall reconsider some of the notions already defined by others. and also introduce others which differ only slightly. New properties (in particular Theorem 1 ) enables us to propose a building algorithm that saves a great number of calculations, and leads to a PERT graph with a minimal number of vertices.
0. Introduction Une des mkthodes pour rboudre un probleme d’ordonnancement est la mCthode PERT [ 1,7]. Le premier pas de cette mithode consiste B construire un graphe (graphe PERT ou amkricain) dont les arcs reprtsentent les tiches et oh les relations de succession sont traduites ainsi: b est successeur de a si et seulement si il existe dans le graphe un chemin dont le premier arc est a et le dernier b. La construction n’est e n gCnCral possible que si l’on introduit de nouvelles tiches, dites virtuelles, de durCe nulle. Kelley [6] note qu’il est avantageux pour rCduire la longueur des calculs suivants de construire un graphe PERT ayant le nombre minimal de sommets. Hayes [5] donne un ensemble de recettes et sa mkthode ne produit pas, en gCnCral, le graphe minimal. Dimsdale [ 3 ] propose un algorithme de construction du graphe PERT minimal. Fisher et al. [4] montrent que l’algorithme de Dimsdale est faux; ils en donnent un nouveau qui est exact; leur article ne contient pas de preuve mathtmatique conformkment au style du journal (CACM). E n [2] Cantor et Dimsdale donnent, avec demonstration, un algorithme exact. Leur mithode qui s’applique a un probleme qui englobe les problkmes d’ordonnancement, gagne en g6ntralitt mais donne lieu, pour notre cas particulier, B des calculs inutilements longs. Syslo [9] cherche 21 minimiser le nombre d’arcs virtuels ce qui est un autre probleme. 147
F. Sterboul, D. Wertheimer
148
Dans le prCsent article notre contribution est la suivante: comme dans [4] nous ne considkrons que le problkme d’ordonnancement. Nous reprenons certaines notions des articles ci-dessus ou nous e n dkfinissons d’autres qui sont voisines. Nous dCmontrons certaines propriCtCs (notamment le ThCorkme 1) qui permettent de donner un algorithme de construction du graphe PERT minimal rCduisant le nombre des calculs.
1. Notations et definitions G = ( X , U ) est un graphe orient6 dont X est l’ensemble des sommets et U celui des arcs. (i, j) arc reliant le sommet i au sommet j. u + u indique qu’il existe un chemin dont le premier arc est u dont le dernier arc est u. Arc redondant: (i, j ) est dit redondant s’il existe un chemin de longueur supCrieure B un de i vers j. i c X on note P ( i ) = { j c X : (j, i)c U}. Q ( i ) = { j c X : (i, j ) c U } , P ( i ) = { j E X : 3 un chemin de j vers i}. Le graphe H = ( Y , V) est dit graphe arc-dual d’un graphe donnC G = ( X , U ) sans circuit et sans arc redondant, s’il existe u n e application f injective X + V telle que: j , i E X : i E P ( j ) ssi f(i) + f(j).
Remarque 1. H ne contient aucun circuit contenant au moins un arc de la forme
fW. Graphe francais [8]. Graphe amtricain. Les donnCes du problkme d’ordonnancement sont reprksentees par le graphe “francais” G = ( X , U ) oh les sommets representent les tAches et ou I’arc (i. j ) appartient a U si et seulement si la tiche i precede immkdiatement la tiche j. Le but du problkme est donc de construire un graphe H arc-dual de G. H est le graphe PERT ou amkricain. On doit minimiser le nombre de ses sommets.
2. Construction
2.1 Graphe Ho A partir du graphe franGais G = ( X , U ) , sans circuit et sans arc redondant, o n construit un graphe H,, = ( Y o ,V(,) de la manikre suivante: pour chaque sommet i E X , on dCfinit deux sommets a, et b,.
Commenf construire un graphe perf minimal
149
Yo= U , , x { a , } U { b , }et V, est constitue des arcs (a,,b,) pour tout i c X ainsi que des arcs (bl,a,) si i E P ( j ) dans G. Les arcs de la forme (a,, b,) seront dits reels et ceux de la forme (bl,a , ) seront dits virtuels. Posons f(i) = (a,,bc). I1 est immkdiat que:
Proposition 1. Ho est graphe arc-dual du graphe G par l'application f. 2 . 2 . Gruphe H , Dans le graphe HO on pose:
ai R ai ssi P ( i )= P ( j ) , bi S bi ssi Q ( i ) = Q ( j ) . On vkrifie aiskment que R et S sont deux relations d'equivalence. Soient a,,. . . , aL, 6,.. . . , b;, les classes correspondantes sur A = U{ai} et B = U { b i } . On appelle contraction de type 1 I'operation qui consiste dans Ho h contracter, en un sommet unique, tuus les sommets d'une mCme classe. On effectue, dans H,, toutes les contractions possibles de type 1 et I'on note HI= ( Y , ,V,) le graphe obtenu. L'application f induit une application de X --+ V, que - I'on continue B noter f. Les sommets de HI seront notes ii,. . . . , aL,b , , . . . , b,.
Proposition 2. HI est graphe arc-dual d u graphe G par l'application f. D'une part les contractions de type 1 n'altkrent pas les relations de succession du graphe H,,, d'autre part f reste une application injective car si, par exemple. f ( i ) et f(j) ont apres contraction mCme origine et mkme extrCmitC o n les considkre comme deux arcs diffkrents. Au contraire si des arcs virtuels se trouvent doubles on ne les prend en compte qu'une seule fois. Un arc (6, iii) est un bon arc de H , si et seulement si:
On appelle contraction de type 2 I'operation qui, dans H I ,consiste a contracter en un sommet unique, les deux sommets d'un bon arc.
Remarque 2.2. (a) Une contraction de type 2 n e supprime aucune relation de succession. (b) On verifie aisement que la definition d'un bon arc (hi, iij) est equivalente B la suivante:
F. Sterhoul. D . Wertheiiiter
150
Theoreme 1. Les bons arcs d e HI n’ont deux a deux aucun sonimer cornmun.
(6,
(a) Montrons que (hi, iil) et a [ )ne peuvent pas etre simultanement deux bons arcs de H I .Comme iii et ii, sont deux sommets distincts de HI on a P ( j )# P(1).
Fig. 1
donc il existe un sommet h de G tel que. par exemple. h E P(1) et h$ P ( j ) . Ceci se traduit dans HI par I’existence de I’arc (&,, a,) alors qu’il n’existe pas d’arc ( G h , aI). Supposons que iil) soit u n bon arc; alors h E P(1) et j~ Q(i) implique h E & j ) . Compte-tenu du fait que hg! P ( j ) , il existe alors dans HI un chemin dont le premier arc est f ( h ) et le dernier f(j) et contenant au moins un autre arc reel f ( e ) tel que e s P ( j ) dans G . Supposons que I’arc (6, ii,) soit aussi un bon arc de H I . alors e E P ( j ) , I E Q(i) implique e E p ( l ) . D’ou. dans H I : f ( e ) f(I); comme f ( h ) - + f ( e ) . on a: f ( h ) + f ( l ) . Ceci implique I’arc ( h . 1 ) est redondant dans G contrairement B I’hypothkse. (b) On montre de meme que deux arcs (6,.ii,) et (hh.Zl) ne peuvent pas ctre simultankment deux bons arcs de H I .
(6,
--+
2.3. Graphe H2
Dans le graphe H , = ( Y , ,V,) on effectue toutes les contractions possibles d e type 2. Soit H 2 = ( Y , , V2) le graphe obtenu. Dans le but de simplifier les notations on continue B noter f I’application de X - V2 induite par f : X --+ V, et les sommets de H2 comme ceux de H , .
Proposition 3. H 2 est graphe arc-dual de G par l’application f: I1 est immkdiat que f reste u n e application injective. Montrons qu’il existe u n
151
Comment sonstruire un graphe pert niininzaf
I
,
f(il) ___t__-’-
-
-
- _- -
---
--
f(j,)
Fig. 2.
chemin de f ( e ) vers f(k) si et seulement si e E p(k): (a) e E p ( k ) implique f ( e )+ f(k): cela rbulte de la Remarque 2.2(a). (b) f ( e ) + f ( k ) implique eE p(k): a cause de la transitivitt de la relation de succession, on peut supposer que les arcs du chemin reliant f ( e ) et f(k) dans H 2 sont tous virtuels. Les sommets intermtdiaires c I . .. . , c, proviennent de la ii],,) de H I . contraction des bons arcs (GI, a],),. . . , (6,,,, Comme (kI,4,)est un bon arc, e E P(jl)et j , Q(i,) ~ implique e E p(j2).Comme (6,?,gI,) est un bon arc, et d’aprks la Remarque 2.2(b), e E p(j2) et j 3 Q(i,) ~ implique e E p(j3).On dkmontre ainsi de proche en proche que e E p(j,,,) jusqu’g m = n et finalement e E &k).
Theoreme 2. Soit H2 = ( Y2,V,) le graphe arc-dual de G = ( X , CJ) par /‘application f construit pricddemment et soit H = ( Y , V) un autre graphe arc-dual par I’application g . Alors il existe Y ’ c Y et une application surjective h de Y’ sur Y2 telle que: i E X auec g(i) = ( a ,0 ) alors f ( i ) = { h ( a ) ,h ( B ) } auec a, p E Y ‘ .
6)
(a) Construisons h : soit i E X avec g ( i ) = (ai,p i ) et f ( i ) = (iii. on pose h ( a i ) = ai et h ( p i )= (b) Montrons que h est bien dtfinie: soit j~ X , j f i. avec g ( j ) = (a;,pi) et f(j)= (ii;, posons h ( q ) = uj et h ( P j ) = (1) Montrons que si a; = ai dans H alors ii, = iiidans H,. I1 faut donc montrer que P ( i ) = P ( j ) dans G. Soit k E P ( i ) . Comme H est arc-dual de G on a g ( k ) + g ( i ) . Comme g ( i > et g ( j ) ont la mCme origine, on a g ( k ) + g ( j ) , d’oh k E P ( j ) dans G. Si on avait k $ P ( j ) , il existerait un sommet I de G tel que 1 E P ( j ) et k E p(1). Alors du fait que 1 E P ( j ) et que g ( i ) et g ( j ) ont la m6me origine on a g(1) + g ( i ) et I E P(i). L‘arc ( k . i ) serait alors redondant dans G. contrairement 2 I’hypothkse. Donc k E P ( j ) et P ( i )= P ( j ) . (2) On montrerait de mkme que si 0; = Oi alors bj = (3) Montrons que si pi = aidans H alors = iij dans H,. Montrons d’abord que pi = a; dans H implique i E P ( j ) dans G. En effet on a g ( i ) + g ( j ) , d’oc i E &). Si on avait i$ P ( j ) , il existerait e tel que e E P ( j ) et i E P ( e ) dans G .
6.
K),
6.
6
6.
E Sterboul. D. Werrheimer
152
i
Fig. 3.
D’oh dans H, g ( i ) + g ( e ) et g ( e ) -+g ( j ) , ce qui entrainerait I’existence dans H d’un circuit contenant l’arc rCel g ( e ) . contrairement B la Remarque 1 . On a donc i E P ( j ) , donc 6, 4 est un arc virtuel de HI. Pour montrer que 6 = 4 dans H2il reste donc B moctrer que (6,iii) est un bon arc de HI. Soit donc k E P ( j ) et 1 E Q(i). Alors g ( k ) + g ( j ) et g ( i ) -+g(/). Comme l’origine de g ( j ) est aussi I’extrgmitC de g ( i ) . on a g ( k ) -+ g(/), d’oh k E p(1).
Fig. 4.
Comment construire un graphe pert minimal
153
Corollaire. Le graphe H2 a le nombre minimal de sornmets parmi tous Ies graphes arc-duaux de G.
3. Algorithme
On dkcrit maintenant les constructions prkckdentes SOUS forme algorithmique. Les Sections 1 , 2 et 3 correspondent B la construction de H,, et H I .La Section 4 correspond B la dktermination des bons arcs. La Section 4.2est facultative, mais permet une Cconomie d’opkrations quand elle vient a Ztre utiliske. L’utilisation de la liste T dam 4.2 et 4.3 permet dexploiter la propriktk dkmontrke dans le ThCorbme 1. La Section 4.5 donne la description finale du graphe H2cherche. On suppose donnC le graphe franGais G dont les arcs redondants ont e t t supprimks (par exemple en utilisant les parties I et I1 de I’algorithme donnk dans [4]).Les sommets de G (les taches) sont reprksentb par les entiers i = 1, . . . , N. Pour tout i on connait les listes P ( i ) , Q ( i ) et P ( i ) .
ALGORH’HME 1.
2.
3.
On dCtermine la partition de 11,. . . , N} en classes C , , . . . , C,: i et j appartiennent B la mCme classe si et seulement si P(i)= P ( j ) . On dresse une liste i l , . . . , ,i constituke d’un reprksentant dans chaque classe. On dktermine la partition (1, . . . , N } en classes D1,. . . , DM:i et j appartiennent B la mZme classe si et seulement si Q(i) = Q ( j ) . On dresse une liste j l . . . . ,j M constitute d’un reprksentant dans chaque classe. Pour tout I, 1 < 1 s L on constitue les listes: ~ ( i ,=) {j,
I jm E ~ ( i , ) 1, s m S M }
et
V(ir)= { j m I jmE P ( i l ) , I s m s M )
et pour tout m, 1s m S M , la liste W(j,) = {if 1 il E Q(jm),1 S I S L } .
4. On pose m = 0. On utilise une liste T, indexCe de 1 B M, vide au dkpart. 4.1. Augmenter m d’une unit&;si m > M aller en 4.5.Si W(j,,,)= g aller en 4.1. Si I W(j,)I = 1, soit {il}= W ( j m ) aller , en 4.4. 4.2.Soit E = (0V(s)), s E W(jm).S’il existe il E W(j,) tel que V ( i , )= E et it ne figure pas dans T, alors aller en 4.4. 4.3.Soit F = s E W(jm). S’il existe il E W(j,) tel que V(i,)= F et il ne figure pas dans T, alors aller en 4.4, sinon aller en 4.1. 4.4.On pose T ( m )= if. Aller en 4.1. 4.5.L‘ensemble des sommets du graphe cherchk H2 est I’ensemble des entiers
(nv(s)),
{2j1,. . . , 2jM} U (24 + 1 1 il ne figure pas dans T, 1 < 1 < L}.
F. Sterboul, D. Werrheirner
154
L’ensemble des arcs de H2 est: arcs riels: pour tout i E (1, , . . ,N}, soient 1 et m tels que i E C, et i E Dm: arc (2i,+ 1, 2jm) si il ne figure pas dans T, arc (2jk,2jm) si il = T ( k ) . arcs uirtuels: pour tout m E (1, . . . , M } et tout i, E W(jm):
arc (2jm,2il + 1) si il ne figure pas dans T, arc (2jm.2jk) si il = T ( k ) ,k # m.
Bibliogaphie [l] C.G. Bigelow, Bibliography on project planning and control by network analysis, Operations Res. 10 ( 5 ) (1962). [2] D.G. Cantor et B. Dimsdale, On direction-preserving maps of graphs, J. Combin. Theory 6 (1969) 165-176. [3] B. Dimsdale, Computer construction of minimal project networks, IBM Systems J. 2 (March 1963) 24-36. [4] A.C. Fisher, J.S. Liebman et G.L. Nemhauser, Computer construction of project networks, Comm. ACM 11 (7) (1968). [ S ] M. Hayes, The role of activity precedence relationships in node-oriented networks. dans: Project Planning by Network Analysis (North-Holland, Amsterdam 1969) 128. [6] J.E. Kelley, Critical path planning and scheduling-mathematical basis, Operations Res. 9 (3) (1961) 296-320. [7] D.G. Malcolm, J.H. Roseboom. C.E. Clark et W. Fazar, Applications of a technique for research and development program evaluation, Operations Res. 10 (6) (1062). [8] B. Roy, Graphes et ordonnancement, Revue Francaise de Recherche OpCr. 25 (4ikme trimestre 1962). [9]M.M. Syslo, Optimal constructions of reversible diagraph, preprint.
Annals of Discrete Mathematics 9 (1980) 155-162 @ North-Holland Publishing Company
DECOMPOSING COMPLETE GRAPHS INTO CYCLES OF LENGTH 2p'" Brian ALSPACH and Badri N. VARMA Department of Mathematics, Simon Fraser University, Bumaby, B.C. V5A lS6, Canada It is shown that the complete graph K,, can be decomposed into edge-disjoint cycles of the same length 2p' if and only if n is odd, n 8 2 p e , and 2p" divides (2") where p is any prime and e is a positive integer.
Let K,, denote the complete graph with n vertices. If E(K,,),the edge-set of K,,, can be partitioned so that each partition set is a cycle of some fixed length r, then we say that we have an isomorphic factorization of K,, into cycles C,. We denote such an isomorphic factorization by C, 1 K,,. An isomorphic factorization of K,, into cycles C, is also often referred to as a decomposition of K,, into edge-disjoint cycles of length r. Since a cycle in a graph contributes either degree 0 or degree 2 to a vertex of the graph, it is clear that C, I K,, implies that n is odd. It also must be the case that n 2 r and that r divides JE(K,,)I= (2"). These three necessary conditions for C, I K,, to hold are also sufficient as far as is presently known. This is the case when r = 2' for e 2 2 which follows from work of Rosa [6] and Kotzig [4]. When r = 3 we have the well-known case of Steiner triple systems. The cases of 5, 7, and 9 have been done by Bermond and Sotteau [2]. The cases for r even and 4 s r s 1 6 have been done by Bermond et al. [ 11. A recent survey article on cycle decompositions is by Bermond and Sotteau [3]. In this paper we give another infinite class of r's for which the necessary conditions are also sufficient. We now state the main theorem and use the rest of the paper to prove it.
Theorem 1. Let p be any prime and e be any positive integer. Then C Z p1=K,, i f and only if n is odd, n 22p', and 2p" divides (2"). If p = 2 , then 2pe=2e+l and the necessary conditions imply that n = m * 2'+*+ 1 for some positive integer m. Kotzig [4] and Rosa [6] together proved k 1 K Z m k + l for any even k 2 4 . Setting k = 2p" proves the theorem in the that c case that p = 2 . *This research was supported by the Natural Sciences and Engineering Research Council of Canada.
155
B. Alspach, B.N. Varma
156
Henceforth, p will always be an odd prime. We now state a result of Sotteau that will be used several times in the rest of the proof.
Theorem 2 (Sotteau [7]). Let k = 2 t be an even integer with k a 4 . Then the complete bipartite graph K , , can be isomorphically factored into cycles C, if and only if r 3 t, s a t, and 2t divides rs. We now examine the arithmetic implications of 2p" dividing (3.There are two cases. Either p' divides n or p' divides n - 1. First we consider the case that p' divides n - 1. Since 2 also divides $(n- l), we know that n = 2m . 2p' + 1 for some positive integer m.We then have CZp= 1 K,, by the Kotzig-Rosa result stated above. This leaves us with the case that p' divides n. In this case it is easy to verify that either n = 4mp" + p' when p" = 1 (mod 4) or n = 4mp' + 3p' when p' = 3 (mod 4). When n = 4mp' + p' we write K,, in the form K n
= K4mp'+p' = K4(rn-l)p'+(5p'-1)+1
= K4(rn-I)p'+l
UK~peUK4(m-l)pe,5p'-l.
The complete graph K 4 ( m - l ) p e + l can be decomposed into cycles C2peby the Kotzig-Rosa result. The complete bipartite graph K 4 ( m - l ) p e , 5 p e - 1 can be decomposed into cycles CZpeby Theorem 2. Thus, we see that Czpe1 K,, if we can decompose Ksp=into cycles CZpe. When n = 4mp' +3p' we write K,, in the form Kn = K 4 m p e + 3 p e - K4mp'+(3p"-1)+1 -K4mpe+1
U K 3 p e - 1 UK4rnpe.3pc-1.
The Kotzig-Rosa result gives a decomposition of the complete graph K4mpe+linto and Theorem 2 gives a decomposition of the complete bipartite graph cycles CZp= K4mpe,3pe-1 into cycles Czp=. Thus, we see that C2pe1 K,, if we can decompose K3pe into cycles C2pe. The problem has now been reduced to finding two particular decompositions. We shall first show that C2p= 1 K3,,e when p3= 3 (mod 4). We do it by proving a more general result which we now state.
Theorem 3. If m > O and m = 3 (mod 4), then C,, I K,,,,. First, partition the vertices of K , , into rn disjoint triples, that is, let V(K,,) = V(K,,) defined by u ( u i i )= u i ,j + l for 1s i s m and j = 1, 2, and 3 where 3 + 1= 1. The cycle { u i j : 1s i s m and j = 1,2, and 3). Let u denote the permutation of
c=
~ l l ~ 2 2 u 3 1 u 4* 2 * . u m 1 u 1 2 ~ 2 1 u 3 2 '
* '
u r ~ - l ,lum2ull
Decomposing complete graphs into cycles of length 2p'
157
has length 2rn. Define a(C) by the action of (T on the vertices of C. The three cycles C, o(C), and a2(C) all have length 2rn, are edge-disjoint, and do not contain any of the edges of the form uijui+l, where the first subscripts are taken modulo rn. Now the set of edges of the form uiiui+l, form three vertex-disjoint cycles of length rn connecting vertices of successively indexed triples. The importance of this is that if we take any permutation of the rn triples and find three vertex-disjoint cycles of length rn connecting successive triples along the permutation, then the remaining edges may be partitioned into three edgedisjoint cycles of length 2 m connecting successive triples along the permutation. We now make this more precise. Let a be a permutation of the set {1,2,. . . , rn}. If the three cycles u a ( 1 ) , j ( l ) [ ~ a ( 2 )j(2)t ,
* * *
urn(,), j(rn)iUa(l), j(l)i
for i = 1, 2, and 3
are vertex-disjoint, then the remaining edges joining triples { u ~ ( ~ u) , ( ~ )2,, u , ( ~ 3)}, to triples { u ~ ( ~ 1,+ ~ ) , 2, 3} for i = 1, 2, . . . , rn (where rn + 1= 1)can be partioned into three edge-disjoint cycles of length 2rn. We now use the well-known decomposition of K,,, n odd and n 2 3 , into edge-disjoint Hamiltonian cycles [ 5 , p. 1611. An example of the construction is given in Fig. 1. The Hamiltonian cycles are obtained by rotating the given cycle through each of the first $(rn- 1) clockwise possible rotations (the identity rotation counts as one possible rotation). Let a be the permutation of { 1,2, . . . , rn} given by the first Hamiltonian cycle in the decomposition of K , into edge-disjoint Hamiltonian cycles, that is, a ( i )is the index of the ith vertex of the cycle starting at the vertex ul. Now let C be the cycle u11u12ua(2),2ua(2),
l U a ( 3 ) ,l U a ( 3 ) , 2 ' '
~ a ( ~ - l ) , 2 ~ a ( m - l ) , 3 ~ a ( m ) , 3l ~ u al l(. ~ ) ,
It has length 2rn. The cycles C, a(C>, and a2(C) all have length 2rn, are edge-disjoint, use all of the internal edges of the triangles formed by the triples, and their edges between successive triangles along the permutation a form three vertex-disjoint cycles of length rn. Hence, the edges not used between successive triangles can be partitioned into three edge-disjoint cycles of length 2rn.
I
/ u1
n
3
u6
Fig. 1.
B.Alspach, B.N. Vama
158
Since m = 3 (mod 4), there are an odd number of Hamiltonian cycles in the decomposition of K, into Hamiltonian cycles. We have used one of the Hamiltonian cycles above to use all the internal edges of the triangles along with the edges connecting the triangles in the order of the Hamiltonian cycle. This leaves an even number of Hamiltonian cycles in the decomposition of K,. We then pair them to complete the decomposition of K3,. We pair the Hamiltonian cycles of the decomposition of K, so that any pair are successive rotations of the cycle of Fig. 1. Thus, it suffices to consider the cycle of Fig. 1 together with the next one in the clockwise rotation of the cycle, that is, the two cycles we consider are
c=
u1u2u3
* * *
U,-1UmU1
and
c' =
u 1 u 3 us u 2 u 7 u 4 u g u 6
* ' *
um-2
urn-5
Urnl&-3
urn-1 u 1 .
These two cycles determine two permutations of the triangles (triples) of K3,. These two permutations tell us how to move among the triangles. Consider the cycle that is displayed in Fig. 2. A solid line in the figure corresponds to an actual
um-l,l
LL
um2
0 m3
--? lJm-1,2 I
8 0
0
Fig. 2.
I
0
9
Decomposing complete graphs into cycles of length 2p"
159
edge of K3,, a broken line corresponds to a path from uii to uki by having the first index change according to the cycle C from i to k in the increasing direction of indices, and the wavy line from ~ 7 to 1 urn+, corresponds to a path from u71 to such that the first indices change according to the index changes following the cycle C' from u7 to where we pass through the vertex u4. On both the broken line and wavy line paths the second indices stay constant. Call the cycle that we obtain from Fig. 2 by C*.First, the cycle C* passes through 1,2, . . . , m as a first index exactly twice. Thus, the length of C* is 2m. Second, if we rotate the Fig. 2 configuration so that the first block goes onto the second block, the second block onto the third block, the third block onto the first block, and keep the second indices fixed, then this rotation and the second power of the rotation give us two more cycles of length 2m. All of the cycles have no edges in common and they follow the edges according to the cycles C and C'. In fact, the two cycles of length 2 m we obtain are nothing more than a(C*) and u2(C").
Now we want to examine the edges of C*,v(C*), and a2(C*)that lie between triples given by C.Notice that the edge ulluzl is present since the edge u13U23 is in C*. Likewise, the edge ~ 2 1 ~ is 3 1present since the edge u23u33 is in C*.In fact, if one examines the diagram of Fig. 2, it will be noticed that every edge of C* that is following the cycle C does not have a change in the second index. Hence, the edges of C*,a(C*), and aZ(C*)contain the three vertex-disjoint cycles D = u11u21u31 * * * u m I u l l , a(D), and a2(D).Therefore, by earlier remarks, the remaining edges of K 3 , that join triples according to the cycle C of K , can be partitioned into three edge-disjoint cycles of length 2m. Now we examine the edges of C*,a(C*),and a2(C*)that lie between triples given by C'. In the diagram given in Fig. 2, there are some edges between triples lying along C' in which there is a shift in the second index between end-vertices of the edges. If one calculates the total shift involved in the cycle C",it totals to a shift of +6. Since we are moving with triples, after we have traversed all of C', we will be back at the same second index at which we started. Thus, the cycle
together with u(D') and a2(D')is contained in the edges of C*, a(C*), and a2iC*).It is then the case that the cycles D', a(D'), and a'(D') are vertexdisjoint so that the remaining edges of K3, that go between triangles that follow the cycle C' in K , can be partitioned into three edge-disjoint cycles of length 2m. We have now seen that it is possible to take two different Hamiltonian cycles among the triples of K3,,, and define three cycles of length 2 m using the Hamiltonian cycles so that the remaining edges between the triples along the Hamiltonian cycles can also be partitioned into cycles of length 2m. This allows us to decompose K , , into cycles of length 2m that are edge-disjoint. This completes the proof of Theorem 3.
B. Alspach. B.N. Varma
160
The only remaining result needed is the following. It is proved similarly to Theorem 3.
Theorem 4. If m > 0 and m = 1 (mod 4), then C2, I K,,. First, partition the vertices of K,, into m disjoint sets of cardinality five. So we have V ( K , ) = {uii:1 =sis m and j = 1 , 2 , 3 , 4 , 5 } . Let (T now denote the permutation of V(K,,) defined by a(z+)= q.i+lfor 1 Ci G m and j = 1 , 2 , 3 , 4 , and 5 where we take 5 + 1 to be 1. The cycles
cl=ullu22u31u42
* ' '
um1u12u21u32
' *
um1w13u21u33
' * ' um-1,1um3u11
*
K m - l , 1um2u11
and c2=u11u2?u31u43 *
* '
both have length 2m. The ten cycles C1,C2,u(C1),a(C2),a2(C1),a2(C2),a3(C1), a3(C2),a4(C1),and a4(C2)are all edge-disjoint, have length 2m, and do not contain any edges of the form uiiui+l,iwhere the first subscript is taken modulo m. Now the set of edges of the form uiiui+l,iform five vertex-disjoint cycles of length rn connecting vertices of successively indexed 5-sets. As before, the importance of this is that if we take any permutation of the 5-sets and find five vertex-disjoint cycles of length m connecting successive 5-sets along the permutation, then the remaining edges may be partitioned into ten edge-disjoint cycles of length 2m connecting successive 5-sets along the permutation. Since we have already done the case of 3m in detail, we outline the rest of the proof in the current case of 5m. Again we use the decomposition of K, into Hamiltonian cycles as given in Fig. 1. Since m = 1 (mod 4), there are an even number of Hamiltonian cycles in the decomposition of K,. Now let a be the permutation of {1,2, . . . , m} given by a Hamiltonian cycle of the decomposition where a ( i ) is the index of the ith vertex of the cycle starting at the vertex ul. Now let C be the cycle u11'12ua(2).2ua(2).
l'a(3),
l'a(3),2
* * '
Ka(m-3),2Ku(m-3),3Ka(~-2),3
~ ~ ~ r n - ~ ~ , ~ ~ a ~ m - ~ ~ 5 u,e ( ~ m )~, 1 a U l~ t. r n - ~ ~ ,
It has length 2m. The cycles C,c+(C),a2(C),a3(C), and a4(C)all have length 2m, are edge-disjoint, use all of the internal edges of the 5-sets that join successively indexed vertices within the 5-sets, and their edges between successive 5-sets along the permutation a form five vertex-disjoint cycles of length m. Hence, the remaining edges between successive 5-sets along the permutation a may be partitioned into ten edge-disjoint cycles of length 2m. We then do the same trick with a second Hamiltonian cycle of K, to use all of the internal edges of the 5-sets whose endvertices differ by two on their indexes. At this point we will have used two Hamiltonian cycles of K, to use all the internal edges of the 5-sets. This leaves an even number of Hamiltonian cycles in the decomposition of K,. As before we pair them and may assume that we are
Decomposing complete graphs into cycles of length 2p'
161
..= Fig. 3.
taking successive Hamiltonian cycles of Fig. 1. We assume the first cycle is u1u2u3 * u , - ~ u , u ~as we also did in the previous proof. Now consider the cycle of Fig. 3. The conventions are the same as for the cycle of Fig. 2. Hence, one can see that the cycle has length 2m and under the appropriate powers of (T we obtain five edge-disjoint cycles of length 2m. By counting the label changes in the second index, it is easy to see that one obtains five vertex-disjoint cycles of length rn along each of the Hamiltonian cycles. Thus, the remaining edges may be partitioned into ten edge-disjoint cycles of length 2m along each of the Hamiltonian cycles of K,. Theorems 3 and 4 together with earlier remarks about recursively constructing cycle decompositions complete the proof of Theorem 1. It is likely that the techniques employed in this paper will prove useful in attacking harder cycle decompositions of complete graphs.
-
References [l] J.-C. Bermond, C. Huang and D. Sotteau, Balanced cycle and circuit designs: even case, Ars Combinatoria 5 (1978) 293-318.
162
B. Alspach, B.N. Varma
[2] J.-C. Bermond and D. Sotteau, Cycle and circuit designs odd case, in: Graphen Theorie und deren Anwendungen, Proc. Int. Colloq. of Oberhof (1977) 11-32. [3] J.-C. Bermond and D. Sotteau, Graph decompositions and G-designs, Proc. Fifth British Combinatorial Conference, Congress. Num. 15, Utilitas Math. (1975) 53-72. [4] A. Kotzig, On the decomposition of complete graphs into 4k-gons, Mat.-Fyz. casopis Sloven. Akad. Vied. 15 (1965) 229-233 (in Russian). [5] E. Lucas, Recreations mathematiques, Vol. I1 (Gauthiers-Villars, Paris, 1883). [6] A. Rosa, On cyclic decompositions of the complete graph into (4m + 2)-gons, Mat.-Fyz. easopis Sloven. Akad. Vied. 16 (1966) 349-353. [7] D. Sotteau, Decomposition of K , , , J K Z , J into circuits of length 2k, J. Combin. Theory (B), to appear.
Annals of Discrete Mathematics 9 (1980) 163-174 @ North-Holland Publishing Company
UNE GENERALISATION DANS LES pGROUPES ABELIENS ELEMENTAIRES, p>2, DES THEOREMES DE H.B. MA" ET J.F. DILLON SUR LES ENSEMBLES A DIFFERENCES DES 2-GROUPES ABELIENS ELEMENTAIRES P. CAMION CNRS et INRZA. 3, rue de Framois Couperin, Domaine de la Bataille, 78370 Plaisir, France
1. Introduction Soit G un groupe abklien. Classiquement, une partie D de G est un ensemble B diffkrences (bribvement E.D.) lorsque Card(D fl D + h ) = A, pour tout h E G \{O}. Le rCsultat ktonnant de H.B. Mann [ll] est que pour un groupe G d'ordre 2", somme directe de groupes d'ordre deux -i.e. G est form6 de tous les mots binaires de longueur k-alors pour un ensemble 2 diffkrences D de G, on a nkcessairement si D n'est pas le complkmentaire d'un sous-groupe de G augment6 de 0:
k
= 2t,
Card D = 2'-'(2'
+E ) ,
h = 2'-'(2'-'+
E),
E
E (1, -1).
Le but de la dkmarche que nous retracons ici et qui est dkveloppke en dCtail dans P. Camion [l], est d'obtenir un rksultat dam les p-groupes abkliens ClCmentaires duquel le thkorbme de Mann serait un cas particulier. Remarquons que le complkmentaire de D dans G est lui aussi un E.D. Ses parambtres s'obtiennent en remplaGant E par - E dans la formule donnCe. C'est dire que les formules de Mann dkterminent sans aucune latitude les valeurs des paramktres. En nous appuyant sur les travaux de divers auteurs, parmi lesquels J.M. Goethals, H. van Tilborg et P. Delsarte apportent les matCriaux essentiels, nous allons devoir dkfinir un objet plus g6nCral que l'ensemble 2 diffkrences classiques pour p 3 2 et nous obtiendrons dans certaines hypothbses (autodualitk) les expressions de ses parambtres dans lesquelles interviendront des variables entibres dont les bornes seront dCterminCes de faGon prkcise. Pour le cas particulier d'un E.D. classique chacun des domaines de ces variables se rkduira 2 un seul point, on aura nkcessairement p = 2 et les expressions des parambtres seront celles de Mann. 163
P. Camion
164
2. Ensembles b differences binaires Des techniques de construction d’ensembles diff Crences binaires et leurs propriCtCs furent CtudiCes par Kesava et Menon [19], Rothaus [13] Dillon [6] et autres auteurs. Donnons un exemple simple de E.D. Soit G=(ff24,+) et D la partie de G formCe des colonnes du tableau que voici (Fig. 1).
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 1 1
Fig. 1
On a d o = C a r d D = 6 , A =2. Puisqu’ici k =4, t = 2 et les formules de Mann donnent en effet
do=21(22-1);
A=2l(2l-l).
A peut Ctre dkfini comme le nombre de diffCrences h - h ‘ d’C1Cments h, h e D produisant chaque ClCment non nu1 de G. Voici un autre exemple pour k impair qui est donc nkccessairement le complCmentaire d’un sous-espace augment6 de 0 (voir Fig. 2); k =3, d o = 7 , A=6.
0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 Fig. 2.
On dCduit aisCment de [l, formule (48)] que cet exemple de E.D. est le seul pour k impair. Si nous supprimons (OOO)T de l’ensemble de ces colonnes, nous obtenons un ensemble D’ tel que
V h E D’, Card{(g, g‘) I (8, 8‘)E D’ X D‘, g - g’ = h } = 4 = A. et V h # 0 , h$D’, C a r d { g , g ’ ) [ ( g , g ‘ ) ~ D ’ X D ’ , g - g ’ = h } = 6 = A , .
C’est ce que nous nommerons un ensemble partiel h diffkrences (brikement E.P.D.); cette notion fut introduite par Chakravarti et Suryanarayana.
165
Sur les rhiorimes de Mann et Dillon
Pour un tel ensemble D’, tout ClCment non nu1 de D’ s’obtient A, par diffkrences dans D’ et tout ClCment non nu1 hors de D’ s’obtient A l fois par differences dans D’. Lorsque A o = A, on retrouve la definition classique. Wolfmann [16] observa qu’un thCorkme de Goethals et Van Tilborg sur les codes 1inCaires pouvait Ctre 6noncC dans le cas binaire en termes de E.P.D.:
Une partie propre D de G = (IF,k, +) est l’ensemble des colonnes d’une matrice gtntrafrice d’un code ayant exactement deux poids non nuls ssi D est un E.P.D. Rappelons ici quelques ClCments de la thCorie des codes. Un mot de code de longueur n est un vecteur du IF,-espace vectoriel IF: (ici q = 2). Un code lintaire (e est le sous-espace vectoriel sur IF, sous-tendu par les lignes d’une matrice B coefficients dans IF, nommke matrice gtntratrice du code. Le poids d’un mot de code est le nombre de ses composantes non nulles. Nous allons faire voir sur un exemple la dualit6 CtudiCe par P. Delsarte des codes ayant deux poids non nuls. La matrice gbnkratrice du code que nous allons examiner sera formCe des colonnes de la Fig. 1, sans le zBro. Ces colonnes forment donc un E.P.D. (voir Fig. 3) que l’on retrouve dans le tableau ci-dessus
LI 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 1 1
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
0 0 0 0 0 0
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
1 1 1 0 1
1 1 0 1 1
1 0 1 1 1
0 1 1 1 1
1 1 1 0 1
1 1 0 1 1
1 0 1 1 1
0 1 1 1 1
1 1 1 1 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 1
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
0 0 1 1 0
0 1 0 1 0
1 0 0 1 0
0 1 1 0 0
1 0 1 0 0
1 1 0 0 0
0 0 0 1 1
0 0 1 0 1
0 1 0 0 1
1 0 0 0 1
Fig. 3.
166
P. Camion
encadre en haut B gauche. C’est donc la matrice gknCratrice d’un code et sous elle apparaissent les mots de ce code de longueur 5 sous forme de vecteurs-lignes. A gauche de chaque mot de code apparait un vecteur de quatre composantes qui correspond B la combinaison linkaire des lignes de la matrice gknkratrice qui produit ce mot de code. Considkrons alors les vecteurs lignes encadrbs en bas B gauche. Ces vecteurs correspondent aux mots de poids 4 du code. La transposke de la matrice qu’ils forment est la matrice gknkratrice d’un nouveau code, qui, on l’observe sur le tableau, a lui aussi exactement deux poids. Voila la dualit6 pour les codes B deux poids. Alors, par le thkorbme que nous venons d’knoncer, les colonnes de la nouvelle matrice gknbratrice forment B leur tour un E.P.D., dual du premier. Dans le cas particulier que nous venons d’examiner, chacun des deux codes est auto-dual, i.e. chaque parambtre (longueur, premier poids, deuxibme poids) de l’un Cgale le parambtre correspondant de l’autre.
3. Ensembles partiels a differences dans les p -g ro up abeliens elementaires 3.1. Remarques
A ce point de l’exposk nous devons faire quelques observations qui serviront de point de dCpart B la gCnkralisation du thkorbme de Mann. -La notion d’ensemble partiel B diffkrences binaire parait plus naturelle que celle de E.D. puisqu’i tous les E.P.D. correspondent tous les codes linkaires ayant exactement deux poids dont les matrices gknkratrices sont formCes de colonnes distinctes. -A chaque E.P.D. correspondent trois autres E.P.D., i.e. son compl6ment et deux autres obtenus par dualitk. -Le thkorbme de Goethals et Van Tilborg exprime la propriCtk de codes B deux poids non nuls sur un corps fini ff, quelconque. Toutefois, tout code B deux poids non nuls considCrC est projectif, ce qui signifie que deux colonnes quelconques de sa matrice gbnkratrice doivent Ctre linkairement indkpendantes. -La dualit6 de Delsarte vaut pour les codes B deux poids non nuls sur tout corps fini ff,. Notons toutefois que nous ne nous intkresserons qu’au cas oii q est un nombre premier, puisque notre but est d’ktudier des propriktks de groupes abCliens Clementaires. Dans une premibe Ctape, nous allons reprendre les rksultats de Goethals et Van Tilborg et Delsarte dans un CnoncC oii apparaitra la notion d‘ensemble partiel B diffkrences. L‘ensemble des colonnes de la matrice gknkratrice d’un code (e est nommCe ensemble des formes coordonntes de ce code. En effet, u € ( e e 3 g € f f ; : u = ( ( h , g))h,,, oii R est l’ensemble des formes coordonnCes de (e.
Sur les thior2rnes de Mann
et
167
Dillon
On voit aisCment que lorsque l2 est l’ensemble des formes coordonnCes d’un code projectif h deux poids non nuls, alors Do =IF$ est lui aussi l’ensemble des formes coordonnCes d’un code h deux poids non nuls: si les deux poids du premier codes sont wb et wT, alors ceux de l’autre sont (p - l)wb et (p - l)wT, not& respectivement wo et w,.
Exemple d’un E.P.D. SUT IF3. Soit D l’ensemble des colonnes de Fig. 4. Si nous formons toutes les differences possible dans 0, nous obtiendrons une fois chaque ClCment de D et trois fois chaque ClCment de iF$\D. Les param2tres de cet E.P.D. sont donc Card D = 4,
Ao= 1,
A, = 3.
1 0 - 1 1 1 -1
0 -1
Fig. 4.
3.2. Le thiorkme de Goethals et Van Tilborg Un E.P.D. de G est dit non trivial lorsqu’il est distinct de $et l de G. Voici comment se trouve alors 6noncC le theoreme de Goethals et Van Tilborg:
Thhreme 1. Soit Do=[F,Doc[F:.Alors Do est un E.P.D. non trivial ssi il est l’ensemble des formes coordonnies d’un code ’& a deux poids non nuls. Le lecteur trouvera une preuve detaillee du thdorkme en [l]. Les paramktres Ao, Al de 1’E.P.D.et les deux poids wo et w1 du code ’& sont lies comme suit. Soit do = Card Do, puis
ro = do- p(p - l)-lw0;
rl = do- p(p - l)-’wl.
Alors A, = do + Tor1,
Al = do+ rorl - ro - rl,
(2)
et rCciproquement, wO=
~ 1 = ( 2 d o - A o +A,+J((AO--A1)*+4(do-
I)-’, Ao)))/2p(p- I)-’.
(3)
On observe que (Ao- A,) = ro+ rl.
(4)
3.3. La dualit6 3.3.1. Rtsu 1tats g t ntra ux Les entiers ro et rl apparaissent dans la dkmonstration du Thdorbme 1 comme des coefficients de Fourier. Si 5 est une racine primitive p bme de l’unitC dans le
P. Camion
168
plan complexe, on a en effet
V h E FL, h # 0
1 5'"
g,
E {To,
rl>.
(5)
gEDo
Lorsqu'on considbre un ensemble partiel i differences Do dans G, quatre E.P.D. se trouvent implicitement dCfinis: (0) U G \ Do = (0) U D , que nous noterons Dl,o-car nous ne considerons que ceux contenant zQo-EE,, dual de Do dCfini par
et son complkmentaire dans G augment6 de zero, soit El,o. Les rksultats de P. Delsarte sur la dualitt se traduisent comme ceci en termes de E.P.D.
Theoreme 2. L'ensemble Eo dijini en (6) est un E.P.D. ayant pour paramatres d&=CardEo,A&, A;, rh=d&-p(p- l)-'w&,r;= db-p(p-l)-'w; oh d~=(pk-do+ro-pkr1)(ro-rJ', A& = d&+ rbr;,
(7)
A; = d&+ rhri- rh- r;,
et r&=(pk-do+ro)(ro-rl)-l, r; = -(do - ro)(ro- rl)-l.
rb et r; sont des entiers et l'on a
Puisque, par les relations (8) (ro - rl)(rh - I;) = pk,
on a (ro-rl)=--Epi,
~~{l,-l}
et Yon voit que selon la valeur de E , ro sera le coefficient de Fourier dkterminant Eo El.0. Autrement dit, changer le signe de E Cchange ro et rl.
Sur les thiorhes de Mann et Dillon
169
Comme nous le constatons dans [l],
C
VhEE,, h f O ,
( ( h * g ) = l - r 0,
geD1.o
V h E El,
C
(11)
C(h*g)= 1 - r1.
g E D >11 .
Les parambtres po et p1de D , qui est aussi un E.P.D. sont, avec Card D , = d , , p o = dl - do+ A,,
p1=
d , - do+ A,.
(12)
I1 est utile de rappeler ici certaines propriCtCs de ces coefficients de Fourier, r,, r,, r;, r',. Les coefficients sont entiers, et au cours de la dCmonstration du ThCorbme 1, nous observons que
(ro=rl)+(Do=$3,r o = r l = O ) . Nous constatons aussi que, ayant nkcessairement A, equivalentes
G do,
on a les assertions
-A1=0, -Do est un sous-espace vectoriel de F I,; - A0 = do, - r, + rl # 0 et r,rl = 0.
En dehors des cas triviaux oh Do = $3 ou bien Do est un sous-espace de FI,: on a donc ror, < 0.
Exemples. (1) Soit Do l'ensemble des car& augment6 du zCro dans une extenI., Alors, IF,& = Do, car IF, c Do, et Do\{O} agissant sion de degrC pair IFPzL sur F comme un groupe multiplicatif sur FI*:, determine deux orbites sur cet ensemble. On en dCduit sans difficultC que Do est un E.P.D. et on calcule ses parambtres do = &p2' + l), A. = f(p2' + 3), A1
= A0 - 1 = $(p2' - l),
0
--1
- A1 + p'); -1
0-2p
t-1
rl = +(l-p'),
w 1 --1 - Z P t - l (P' + l)(P - 1).
(pt-l)(p-l);
Cet E.P.D. est donc autodual, c'est-&-dire: d6 = do,
r:, = r,,
r', = r,;
par (7) et (8), et en consCquence
A6 = A,,
A', = Al.
P. Camion
170
(2) L’exemple suivant nous montre que pour p et k donnds (G = f f p k , +), un E.P.D. a des parametres qui peuvent prendre plusieurs valeurs. D’autre part, sa construction va faire appel B la rCciproque du ThCorkme 1, i.e. nous construisons un code qui a de faGon Cvidente deux poids non nuls. Voici sa matrice gCn6ratrice
oii j s p . Les deux poids du code sont de facon Cvidente w1 = j(p - l),
wo = ( j- l)(p - 1).
(15)
Do Ctant l’ensemble des colonnes de la matrice donnCe par la Fig. 5, on calcule alors par (1) et (2), do= j ( p - l ) + l ,
rl
ro= p- j + 1,
= 1- j ,
ho= j 2 - 3 j + p + 2 ,
ro- rl = p,
A l = j(j-1).
Cet E.P.D. est lui aussi autodual pour j = 1, . . . , p. 3.3.2. Nouvelles relations entre les paramttres L‘un des rCsultats de P. Delsarte, voir J. Wolfmann [17], est le suivant
Thikreme 3. Soit Do un E.P.D. et w t et wf avec w t < wT les deux poids du code projectif qu’il dktermine, alors
w t = upr
et
wT = ( u + 1 ) ~ ‘ .
(17)
Preuve. On obtient en effet des relations (8) (do- ro) = -r;(ro- rl) et (do-rl)=do-ro+ro-r,. D’oii, par (1) et (lo)’, w$= E p i - l rlI et
w t = epi-’(r; - 1).
D’autre part, p(w$- wT)=rl-ro=&pi,
(10)‘’
171
Sur les the‘orimes de Mann et Dillon
et
(w;-
WT)E
>o.
Donc
u = -r;
lorsque E = -1 (alors r;
et u = rl, - 1 lorsque
E =
1.
Nous obtenons en [l] un resultat qui precise les valeurs que peuvent prendre les paramktres d’un E.P.D. Notons v,(j) la valuation p-adique de l’entier j . On a le
Thhreme 4. Soit i = u , ( r o - r l ) = u p ( w , - w o ) + l . vp(Al) = s d i. Notant 1 l’entier k - 2i + s, on a nicessairement A l = psy’(ply’- E ) ;
Si do
et 2 i s k , alors
p ( p - l)-l(w0+w l ) = 2do- ro- rl = pi(2y’p*- E )
(18)
pour un entier y’, (y’,p) = 1, et 1 s y’<+i+l-s, et E ~ ( 1-1). , Si s = i, alors E = 1 . Dans ce cas le E.P.D. est nicessairement le compliment duns G d’un sous-espace sur F, de IFp* augment6 de z6ro. Si p = 2 et s = i - 1 , alors y ‘ = 1. y ’ = 1 et
Les exemples donnCs permettent de vkrifier que les inkgalit& obtenues sont les meilleures possibles. I1 y a des exemples oh s = 0 ou s = i - 1 (s = i est donc trivial), des exemples pour lesquels y’ = 1 ou bien oh y‘ prend les plus grandes valeurs permises. Redonnons une preuve du lemme de [l] qui conduit aux gknkralisations des thkorkmes de Mann et Dillon.
Theoreme 5. Lorsque 2i d k , alors avec ro- rl = -&pi, P-l(do- ro) =
wt=
y’pk-i+s-l
;
I
k-2i+s
rl,=eyp
Preuve. Le ThCorkme 4 nous assure en effet, lorsque 2 i S k , que p(w$+ wT) = pi(2y’p’ - E ) ,
(20)
avec ( y ’ , p ) = l et I = k - 2 i + s . L‘assertion dkcoule alors de (10)” et de la relation preckdente.
3.3.3. L’autodualiti 3.3.3.1. Dijinition Donnons une definition precise de l’autodualit6 en utilisant les notations introduites au dkbut de 3.3.1.
P. Camion
172
Definition. Do est autodual lorsque lui-m&me ou Dl,o a les mCmes pararnbtres que Eo. Propriete 1. Do est autodual ssi l'une des deux assertions que voici est vraie: (i) Card Eo = db = do = Card Do, rb = ro, r; = rl. (ii) d; = pk - do+ 1, rl, = 1- rl, r; = 1- ro.
Preuve. Dans le cas oh Do a les rn&mesparambtres que Eo on peut identifier les coefficients de Fourier de ces deux ensembles et puisque (ro- rl)(rl,- r ; ) > O , on a bien (i). RCciproquement, par le ThCorbme 1, (i) nous assure que Do a les rnCmes parambtres Ao, A,, wo, wl,respectivernent que Eo. 3.3.3.2. L'autodualiti et les thdorimes de Mann et Dillon Reprenons d'abord un rksultat dCmontrC par J. Wolfmann et que nous pouvons aisCment indrer ici
Thhreme 6. Do est un E.P.D. autodual ssi
g ( r o - rl) = vp(wl - wo)+ 1 = 4k. Preuve. La condition est nCcessaire, par la PropriCtC 1, puisqu'alors rb - r; = ro - r1 et que nous avons toujours (ro - rl)(r; - r;) = pk. Soit alors k = 2t. Par le ThCorbrne 5 , ro- r, = -EP' et r; = ~y'p'. Ensuite, du ThCorbme 4 et des relations (2) do- ro+ rl(ro- 1)= Al = psy'(psy'-
E).
(21)
Par (lo') et (8) rl(rl - EP' - 1)= r;(r; - EP' - 1).
(22)
En rCsolvant 1'Cquation du second degrC en r;, on obtient r;= r, ou r ; = -r,+Ept+l=l-ro. Par la propriCtC 1, nous devons donc obtenir respectivement d & = d o ou d&=pk-do+l. On obtient en effet dans le premier cas, par le ThCorbme 2, (7) et (€9, db - r;
= Ep'rl = Eptr; = do- r,,
(23)
et dans le second cas d;-l+rl=d;-r;,=Eptrl= = pk
~p'(l-r;)= ~ p ' ( l - r ; + ~ p ' )
+ ~ p ' -Ep'r; = pk + ~ p ' -do+ ro= pk -do+ rl.
3.3.4. Les thiorkmes de Mann et Dillon Nous allons prCsent dkmontrer un thCorbme un peu plus gCnCral que [ l , ThCorbme 81 qui correspond au ThCorbme de Mann pour p >2.
Sur les thkor2mes de Mann et Dillon
173
Thbreme 7. Soit Do un E.P.D. autodual non trivial de IF;' (donc do = d& ro = rb, rl = r;). On a rl = ey'pS, do = Epf(rl- 1) + rl, ho= r:+ rl- ~ p ' ,
(24) hl = r:- rlr
oic s = v,(A), l=Sy'
oiI s = u,(A,). Ensuite, les valeurs de ho et de A l se d6duisent du ThCorbme 4 (18) puisque 2 i = 2 t = k , et de A o - h l = r o + r l (4). Pour le calcul de do d'autre part, on sait par (23) que do - rl + EP' = do - ro = Ep'r; = Ep'rl.
Le fait que y'
Thbreme 8. Soit Do = IFpDoun E.P.D. autodual extrimal de G = (IFp*, +). (Donc k =2t, U ~ ( W ~ - W ~ ) = ~ - ~ = U ~ ( ANotons ~ ) ) . Eo le E.P.D. dual de Do. Soit S un sous-espace sur IF, de F,z, de dimension t - i , - t = S i = S t . Alors, en notant S'- le
174
P. Camion
compldmentaire orthogonal de S dans l’espace vectoriel Card E o nS’ oh
E E
[FPz1,
on a
+ &pi Card D o nS = y ’ p r - l ( p i+ E ) ,
(1, -l}, 1s 7 ’ sp - 1, y’ dtant ditermint par la donnde de Do.
Bibliographie [l] P. Camion, Difference Sets in Elementary Abelian Groups, Skminaire de mathbmatiques
supkrieures (Les presses de I’universitb de Montrbal, 1979). [2] P. Delsarte et J. M. Goethals, Irreducible binary cyclic codes of even dimensions, Univ. of North Carolina, Dept. Statist. Mimeographed Ser. No. 60027 (1970). [3] P. Delsarte, Two-weight linear codes and strongly regular graphs, MBLE Research Laboratory Rep. R. 160 (1971). [4] P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math. 3 (1972) 47-64. [ 5 ] P. Delsarte, Four fundamental parameters of a code and their cornbinatorial significance, Information and Control 23 (1973) 407-438. [6] J.F. Dillon, Elementary Hadamard difference sets, Ph.D. Thesis, Univ. of Maryland (1974). [7] J.M. Goethals et M. van Tilborg, Uniformly packed codes, Philips Res. Reports 30 (1975) 9-36. [S] F.S. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J. 42 (1963) 79-94. [9] F.S. Mac Williams and N.J.A. Sloane, The Theory of Error Correcting Codes (North-Holland, Amsterdam, 1977). [lo] F.J. MacWilliams, N.J.A. Sloane et J.G. Thompson, Good selfdual codes exist, Discrete Math. 3 (1972) 153-162. [ll] H.B. Mann, Difference sets in elementary abelian groups, Illinois J. Math. (1965) 212-219. [12] H.B. Mann, Addition Theorems (Interscience, New York, 1965). [13] O.S. Rothaus, On “Bent” functions, J. Combin. Theory 20(A) (1976) 300-305. [141 H. van Tilborg, Uniformly packed codes, Proefschrift, Technische Hogeschool Eindhoven (1976). [15] J. Wolfmann, Formes quadratiques et codes ?I deux poids, C.R. Acad. Sci. Paris 281 (1975). [16] J. Wolfmann, Propribtbs combinatoires dbduites de l’btude des codes ?I deux poids, C.R. Acad. Sci. Paris 284 (1977). [17] J. Wolfmann, Codes projectifs h deux poids, “caps” complets et ensembles de diffbrences, J. Combin. Theory 23(A) (1977) 208-222. [18] J. Wolfmann, Aspects gkombtriques et combinatoires de I’Ctude des codes correcteurs, Thltse, Universitb de Paris 7, Paris (1978). [19] Kesava et Menon (1960).
Annals of Discrete Mathematics 9 (1980) 175 @ North-Holland Publishing Company.
ON CORRELATIONS OF FINITE BOOLEAN LATTICES D. SCHWEIGERT Universitiit Kaiserslautern, West Germany
Abstract A correlation a of a lattice (I.;A , v ) is a bijective mapping such that a ( x A y) = a ( x ) v a ( y ) and a(x v y) = a ( x ) A a ( y ) . If L is a Boolean lattice with n atoms then the number of non-isomorphic correlations is the number of partitions of n.
175
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Annals of Discrete Mathematics 9 (1980)177-179 @ North-Holland Publishing Company
PROPERTIES OF (0, 1)-MATRICES WITH FORBIDDEN CONFIGURATIONS Richard P. ANSTEE California Institute of Technology, Pasadena, CA 91 125, U.S.A.
We start by giving a precise meaning to the term configuration. A configuration is an equivalence class of matrices where two matrices represent the same configuration if one of the matrices is a row and column permutation of the other matrix. A matrix contains a configuration if a submatrix of A represents that configuration. We wish to study (0, 1)-matrices which do not contain certain configurations. For an arbitrary k (k 2 3) we define an interesting set of configurations Lk. Consider the set of (0, 1)-matrices A with k rows and AAT>O such that any submatrix A’ of A obtained by deleting columns does not have A’(A’)T>O. Here, AT denotes the transpose of A. Let Lk be the set of the associated configurations. Let Jk.l be the k X 1 matrix of all 1’s and identify it with its associated configuration. We have that Jk.1 E Lk. The following result holds.
Theorem 1.1. Let A be a (0, 1)-matrix of size m x n with AAT>O and A contains no configuration in Lk\Jk,l for some k ( m 3 k 3 3 ) . Then A has a column of m 1’s. We note that L3\J3,1consists of one 3 x 3 configuration with row and column sums 2. We call this configuration a triangle. Ryser proved Theorem 1.1 for k = 3 in [2]. Note that if A has no configurations in Lk\Jk,l, then A has no configurations in Lk+l\Jk+l,l. Thus the condition “no configurations in Lk \ & I ) ’ becomes weaker as k gets larger. From Theorem 1.1, we obtain a result about the class C ( S ) defined as follows. Let S be a symmetric positive integral matrix of order m with zero trace. Then C ( S )= {A (0, 1)-matrix I AAT= S + 0, D arbitrary diagonal matrix}. (1.1)
To avoid trivialities we also require matrices in C ( S ) to have column sums greater than 1. This is discussed in [3] and corresponds to looking at collections of m sets with the sizes of the set intersections specified in S. We can prove the following:
Theorem 1.2. For every matrix A E C(S) containing no configurations in L k \ J k , l , then the columns with column sum greater than or equal to k are unique apart from order. 177
178
R.P. Anstee
This is a generalization of a result proved by Ryser [3]. We will define a stronger condition that matrices can satisfy. Let CONDITION(k) be the condition: “contains no configuration in Lk\Jk,l and contains no configuration in {[CD]I C E L ~DEL,\J~,,} , in columns with column sums less than or equal to i (3=si < k)”. The following lemma can be used to derive Theorem 1.4 using matrix techniques developed by Ryser [4].
Lemma 1.3. Let A be a (0, 1)-matrix satisfying CONDITION(k) for some k (k 3 3). Let A,, A, E C ( S )for some S # 0 where Al is obtained as the repetition of certain columns in A and A, is obtained as the repetition of a disjoint set of columns in A. Then A, and A, are equal apart from a column permutation. Let the row intersection of row i and row j ( i f j), regarded as a vector, have a 1 in a given column if both row i and row j do and a 0 otherwise. The following theorem is perhaps our most interesting result.
Theorem 1.4. Let A be a (0, 1)-matrix satisfying CONDITION(k) for some k ( k s 3 ) with column sums greater than 1 . Then the number of linearly indcpendent row intersections is equal to the number of distiitct columns. There are a number of consequences to Theorem 1.4. We have Ryser’s result that if A is of size m X n and has distinct columns in addition to the properties given above then n S ( Y ) (see [4]).Let K , be a (0, 1)-matrix of size m X (2”) with column sums 2 and distinct columns. We order the columns such that if column i has 1’s in rows j and k and column p has 1’s in rows q and r, then i < p if j < q or j = q and k < r. We obtain the following result using the theory of SDRs.
Theorem 1.5. Let A be a (0, 1)-matrix of size m X (2”) satisfying CONDITION(k) for some k (k 3 3 ) with distinct columns and all column sums greater than 1. Then there exists a permutation matrix P of order (2”) such that A s K , P . We note that CONDITION(3) is merely the condition “contains no triangles”. We define a solution (of size m ) t o be a (0, 1)-matrix of size m x (2”) containing n o triangles with distinct columns and all column sums greater than 1. W e have been able to determine a great deal about the structure of solutions.
Theorem 1.6. Let A be a solution of size m. Then the number of columns of column sum l in A for 2 c l s m is m - I + l . One can show that the columns of column sum 1 form an analogue of an
( I - 1)-tree following the definition of Beineke and Pippert [ l ] .It turns out that the matrix P in Theorem 1.5 is unique for a given solution. Perhaps the nicest result is the following theorem which tells a great deal about the inductive buildup of solutions.
Properties of (0, 1)-matriceswith forbidden configurations
179
Theorem 1.7. In a solution A of size m there are two rows of row sum rn - 1. After suitable row and column permutations we have I
0
11
I.
In (1.2) A' is a solution of size m - 1.
An algorithm exists for generating all solutions which utilizes the inductive structure of k-trees. Detailed proofs of the above results will be published elsewhere. A discussion of solutions including Theorems 1.6 and 1.7 appear in [ 5 ] .
References [l] L.W. Beineke and R.E. Pippert, The number of labelled k-dimensional trees, J. Combin. Theory 6 (1969) 200-205. [2] H.J. Ryser, Combinatorial configurations, SIAM J. Appl. Math. 17 (1969) 593-602. [3] H.J. Ryser, Intersection properties of finite sets, J. Combin. Theory 14 (1973) 79-92. [4] H.J. Ryser, A fundamental matrix equation for finite sets, Proc. Am. Math. SOC.34 (1972) 332-336. [5] R.P. Anstee, Properties of (0, 1)-matrices with no triangles, J. Combinatorial Theory (Ser. A) 29 (1980) 186-198.
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Annals of Discrete Mathematics 9 (1980)181-182 @ North-Holland Publishing Company.
ON ADJACENCY MATRICES FOR HYPERGRAPHS Cyriel van NUFFELEN UFSIA, Antwerpen, Belgium
Abstract First, we define a weighted direct hypergraph which is a generalization of the usual non-directed hypergiaph. Second, we introduce adjacency matrices (formerly only used for graphs) for these weighted directed hypergraphs. Third, with an appropriate definition of matrix multiplication we are able to enumerate different kinds of chains in weighted directed hypergraphs.
PRIME TRIANGULAR MATRICES OF INTEGERS J.S. BYRNES University of Massachusetts, Sharen, M A 02067, USA
Abstract Consider lower triangular n x n matrices of integers, which will be represented by capital letters. Let I be identity, and call A,, . . . , A,,, relatively prime on the right (similarly on the left) if there exist XI, . . . ,X,,, such that A I X , + . * + A,X, = I. Call A prime on the right (similarly, on the left) if A is relatively prime on the right to p l for all primes p except for one. Call A a uniprime if its determinant is f a prime. Let the cross-trace of two matrices A = (qi) and B = (b,) be
-
M A , B)= ( a i l , hi)+
* * *
+ (%n,
bnn).
Let A divide B on the right, A/& (similarly, on the left, A/,B), if there exists Q such that B = QA. Let A divide B, AIB, if A/,B or A/,B. Let R be a prime if R/AB implies RA or RIB. Then:
Theorem I. A and B are relatively prime on the right iff ctr(A, B) = n iff A and B are relatively prime on the left. 181
182
A bsrracts
Theorem 11. A is prime on the right iff det A = p k for some prime p and positive integer k iff A is prime on the left. Theorem 111. If there are Q, R, and S with det S # O , +1 such that A = SQ, B = SR, then A and B are not relatively prime on the right, but the converse is false. Theorem IV. If det A# 0 , f1, then A can be expressed as the product of uniprimes. Theorem V. For n > 2, R = (rii) is a prime iff R is a uniprime with r, n-1. For n = 2, R is a prime iff R is a uniprime.
= f1 , 2 S j =z
Annals of Discrete Mathematics 9 (1980) 183-187 @ North-Holland Publishing Company
UNE GENERALISATION D’UN THEOREME DE GOETHALS-VAN TILBORG B. COURTEAU, G. FOURNIER et R. FOURNIER” Uniuersitk de Sherbrooke, Sherbrooke, Quk. J1 K 2R 1, Canada
Camion [l] a introduit et montrC le parti que l’on pouvait tirer de la transformke de Fourier finie dans la construction et 1’Ctude d’une classe d’ensembles B diffkrences. I1 a donnC, en particulier, une caractkrisation d’un code de Hamming comme l’orthogonal d’un code i un poids non-nu1 et une version d’un thtorbme de Goethals et van Tilborg [4] l’effet que DcF,k est un ensemble 2 diffCrences B deux parambtres si et seulement si D est l’ensemble des formes coordonn6es d’un code a deux poids non-nuls. Dans ce travail, nous prksentons un thCor&me contenant les deux rCsultats prCcCdents comme cas particuliers et permettant de mettre en Cvidence certaines propriCtCs combinatoires des codes B N poids non-nuls. Nous isolerons certaines relations explicites entre les parambtres impliquCs qui permettent de retrouver, dans le cas particulier N = 2, les relations de Camion. Dans le cas N = 3, nous trouvons un exemple de ce qu’on pourrait appeler un ensemble B sommes triples. Finalement, utilisant une remarque de Wolfmann [5] et la notion de partition linkaire [2], on montre comment construire des codes B N poids non-nuls. Cette note ne contient pas de preuves et le lecteur est priC de se rkfCrer [3].
1. Codes a N poids non-nuls DCsignons par F, le corps fini a p ClCments et par G l’espace vectoriel F l ;. Soit H une matrice gCnCratrice d’un (n,k)-code IinCaire C. L‘ensemble des formes 1inCaires sur IF; dCfinies par les colonnes de H sera not6 D et appelC l’ensemble des formes coordonnCes du code C (associe a H). Remarquons que n =card D. On utilisera la notation polynomiale pour exprimer les ClCments de l’algbbre du groupe G sur les nombres complexes. Dans cette algbbre, le produit est donne par la formule suivante:
*Cette recherche a Bte supportiie en partie par une subvention du CRSNGC dans le cadre du dBveloppement rkgional, et par des subventions pour dtpenses courantes du CRSNGC. 183
B. Courteau et al.
184 OU
2 x,x~,
x=
y=
gsG
1 y ,x g
et x,, Y , E C .
K ~ G
Dans cette note, tous les ensembles de formes coordonnkes D vkrifient la propriktk lF,D = D. X g dCsigne la fonction caractkristique de D, notons xjh les coeffiSi x = cients de x' dam l'algkbre du groupe G i.e.
xi
=x
.x = 2 XihX"
*
f
u
heG
On dbmontre, par rkcurrence, la proposition suivante qui gbnbralise un rbsultat de Camion [l].
Proposition 1.1. Pour tout h E IF: et pour tout j
=
1,2, . . . , on a
On prouve ensuite le rksultat suivant qui s'avkre trks important dans la preuve de la caractbrisation des codes h N poids non-nuls. Lemme 1.2. S'il existe c I ,. . . , c,, b E Z ou c,# 0 tel que pour tout h EIF: ClXlh
alors le code
-k
V
*
.
*
+ c,X,h
\ {0},
= b,
admet au plus s poids non-nuls.
La preuve de ce rksultat consiste essentiellement h prendre de deux faGons diffbrentes la transformbe de Fourier de
Thboreme 1.3. Soit D l'ensemble des formes coordonndes d'un (n, k)-code lindaire
V sur IF, oii n = card D, vdrifiant la condition F,D = D. Les deux conditions suivantes sont dquivalentes: (i) (e admet exactement N poids non-nuls, (ii) N est dgal au minimum des s tels qu'il existe cl, . . . , c,, b EZ,c, f 0 tels que pour tout h E lFE \ {0}, on ait ClXlh f
* * *
f C,X,h = b.
La preuve de ce thkorbme utilise la transformke de Fourier finie et des calculs sur les matrices de Vandermonde.
185
Une ghniralisation d’un theorhe de Goethals-uan Tilborg
2. Relations explicites et cas particuliers du thhreme Soit % un code 21 N poids non-nuls wl,. . . , wN, dkfini par l’ensemble des formes coordonnQes D ou F,D = D, n = card D. Posons ri = n - p ( p - 1)-’wi, pour tout i = 1, . . . , N. Au cours de la preuve du ThQorbme 1.3, on rencontre les deux relations suivantes:
(L)
f: (-
pk
l)j(
j=l
c
i t < . . . CiN-,
ri, . * . riN-)%h =
(Ti-
n)
j=1
pour h f 0, et
(L)
pkf: j=1
(-I)’(
c
i l < . . . iN-,
ril.
* *
ri,,)xjO=n i C j (rj-ri)[fjj = 1
(rj-n)-pk
fi
j = 1 rj]
qui restreignent Qvidemment les valeurs possibles des parambtres n, rl, . . . , r,, p, k et une discussion comme celle de [11pourrait &re entreprise dans le cas N = 3 par exemple. Des relations prQcCdentes, on peut dQduire facilement, dans le cas N = 1, une caractQrisation d’un code de Hamming comme l’orthogonal d’un code Zi un seul poids non-nu1 et, dans le cas N = 2, on retrouve le thkorbme de Goethals-van Tilborg. Dans le cas N = 3, on peut dkmontrer la proposition suivante en partitionnant adkquatement 5;.
Proposition 2.1. Si D = F;, F,D = D oB D est l’ensemble des formes coordonntes d’un code lintaire % iZ 3 poids non-nuls wl, w,, w3 et si wl+w,+ w 3 = 3np-’(p - l), ou n = card D, alors
est une constante pour tout h E D \ {0}, et
est une constante pour tout h# D.
3. Construction de codes a N poids non-nnls Soit D = 5; l’ensemble des formes coordonnkes d’un code linQaire %. Pour tout h E FF,“\ {0}, notons Hh
I
= {g E F$ (8,
h ) = 0)
l’hyperplan orthogonal Zi h dans Fqk. Remarquons avec Wolfmann [ 5 ] que si a ( h )
B. Courteau et al.
186
est le mot de ‘G: correspondant A h alors son poids est donnC par
w ( a ( h ) )= card D -card@ nH,,). Rappelons Cgalement [2] que P = { Vi I i E I } est une partition linkaire de rang r de Ft si (i) pour tout i E I, Vi est un sous-espace vectoriel de Ft de dimension r oii r divise k, (ii) pour tout if j , Vi n V, = {0}, (iii) 5 : = U{V, I i E I},et (iv) pour tous i, j , l E I, on a que V, n (V, + V,) # ( 0 ) entraine que Vi c V, + V,. On a alors le rCsultat suivant:
Proposition 3.1. Soir P une partition lintaire de rang r de F r et soit Pl= {V, 1 i = 1,.. . , rn} une partie de P utrijiant la condition
-
(C)
pour toute suite V,,, . . . ,V, telle que la sornrne Vi,+ soit directe, alors card{V E Pl 1 V c V,,@. * -03 V,J = 1, est une constante.
+ V,
Alors D = U{V, 1 i = 1, . . . , rn} est l’ensemble des formes coordonnCes d’un code admettant au plus N poids non-nuls. Les poids de ce code se retrouvent dans l’ensemble suivant:
{wl=(rn-lN-1)qr-’(q-l), . . . wj 9
~j
= (rn - lN-j)qr-’(q - I), . . . , W N = rnq‘-l(q - I)}.
4. Un exemple
-
Nous terminons maintenant en donnant un exemple dans le cas N = 3. Soit c$ :LFg F$ la bijection F,-linCaire induite par l’identification Cvidente de IF: avec F22et soit Ed l’ensemble de toutes les droites de IF$. On voit aisCment que +-‘(Ed) = P est partition linCaire de f: en sous-espaces de dimension 2. ConsidCrons P , = {V,, V,, V3, V,, V,} c P oii
v1= {(looooo),(oloooo),(110000), (000000)}, v, = {(001000),(oooloo), (001100), (000000)}, v3 = {(000010), (oooool), (00001l),(000000)}, v, = {( 101010),(010101), (11111l),(OooooO)}, v, ={(100111), (011110),(111001), (000000)}. On voit aisCment, par des considkrations gComCtriques, que P1vtrifie la condition (C) avec I , = 1, I, = 2 (les autres paramktres sont N = 3, r = 2, q = 2, rn = 5 ) . Le code ’G: dont l’ensemble des formes coordonnCes est D = U{V, I i = 1,. . . ,5} admet les trois poids non-nuls w1= 6, w 2 = 8, w g= 10 et on constate que la somme des poids vCrifie la condition de la Proposition 2.1. Appliquant cette proposition, on trouve pl = 76 et p2 = 60.
Une gtniralisarion d’un thtorkme de Goethals-uan Tilborg
187
Bibliographie [l] P. Camion, Difference Sets in Elementary Abelian Groups (Les Presses de I’Universitb de MontrCal, MontrCal, QuC., 1979). [2] J. Constantin et B. Courteau, Partitions linbaires arguesiennes d’un espace vectoriel, Discrete Math., B paraitre,. [3] B. Courteau, G. Fournier et R. Fournier, A characterization of N-weight projective codes, IEEE Trans. Information Theory, soumis pour publication. [4] J.M. Goethals et H. van Tilborg, Uniformly packed codes, Philips Res. Repts. 30(1975) 9-36. [5] J. Wolfmann, Codes projectifs ti deux poids, “caps” complets et ensembles de diffbrences, J. Combin. Theory 23 (A) (1977) 208-222. [6] J. Wolfmann. Aspects gkomktriques et combinatoires de I’etude des codes correcteurs, Thbse, Universitb de Paris 7, Paris (1978).
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Annals of Discrete Mathematics 9 (1980)189-194 0 North-Holland Publishing Company
THE JUMP NUMBER OF DAGS AND POSETS: AN INTRODUCTION
M. CHEIN and M. HABIB C.N.R.S., Structure de l’hformation, Tour 4.5, 4, Place Jussieu, 7.5230,Paris Cedex 05,France
1. Definitions and some related problems 1.1.
In the following, by a dag without further specification, we will mean a finite directed acyclic graph (without directed cycle, but loops are allowed). Let G = (X, U ) be a dag, with 1x1= n, we say that T a total order on X, is compatible with G, if xy E U implies x <7 y. Let us denote by O(G) the set of all total orders compatible with G. Suppose T E O(G), T = x1 x2 * *
I
I
V T E ~ ( G )u,( ~ , G ) + p ( ~ , G ) = n - l .
a ( G )+ p(G)= ~
1 -1.
The study of the jump number was first proposed by J. Kuntzann.
Example.
G
0 (GI
d ~G ),
abcde abced acbde acbed acebd
2 2 2 3 1
P(T,
G)
2 2 2 1 3
The search for the jump number of a dag can be viewed as a problem on posets since we have: for any dags, G1, Gz having the same transitive closure: VT E O(G,)(=O(G,)) U(T, GI)= U(T, G2). 189
(3)
M.Chein, M.Habib
190
Furthermore, in the context of poset theory, the jump number is a kind of measure between a given poset and its “nearest” linear extension. 1.2.
More precisely, M. Barbut in [l], set the following problem: “For a given binary relation R on a set X , find the nearest total order on X , with respect to the distance between binary relations yielded by the symmetric difference (denoted
i-). The jump number problem is a special case of Barbut’s, since for a dag G = (X, U ) if we denote Gh= (X, U,,)its Hasse graph (without transitivity arcs or loops): V T E 0( G )
I Th iGh 1 = 2
G )+ Uh1 - 1 x1+ 1.
U (7,
I
(4)
Furthermore one can give a more general definition of the jump number. Let E be a set (finite or not), and p a partial order on E. Let x, y be two incomparable elements of (E, p), we define the elementary extension p,, of p, as follows: @,,t iff (zpt or (zpx and ypt)). It is well-known, since the Szpilrajn’s paper [111, that p,, is a partial order on E which contains p. So we may define:
u(p)the jump number of p, to be the minimum cardinality of a set of successive elementary extensions needed to obtain a total order ( 5 ) compatible with p. 1.3.
We will now examine the relationship between the path number and the jump number of a dag. A set of elementary paths of a dag G that partitions the vertices of G is called a path-partition of G. Let us denote by p ( G ) the minimum cardinality of such a partition. Trivially we have: For any dag G, p ( G ) - l s u ( G ) .
(6)
Besides, it is well-known that p ( G ) - 1 is the minimum number of arcs that are to be added to G in order to create a hamiltonian path. Similarly, the jump number can be formulated as a completion problem:
a ( G )= min{)VI I G’ = ( X , U U V) is a hamiltonian dag}. By (3) we notice that the arcs of V can be taken as incomparability arcs.
(7)
The jump number of dags and posets
191
Example.
1
2
3
p(G) = 3, as {14,26,35} is a path-partition of G, but in this case we cannot construct a total order by ordering the paths of any optimum path-partition of G. ~ = 1 2 4 3 5 6 a, ( G ) = a ( ~G, ) = 3 > p ( G ) - 1 = 2 . Then it follows: a(G) = min{(P(I P is a path-partition of G and G/P is a dag}- 1.
(8)
Hence, when G is a tree, a ( G ) = p(G)- 1 (let us recall that for any dag G, p(G) is quite easy to compute as we can use any maximum matching algorithm on the incidence bipartite graph). So in conclusion: For any dag G dim(G')- l a a ( G ' ) - 1 = p(G')- 1 Gp(G)-lSp(Gh)-lSa(G).
(9)
(Where G' denote the transitive closure of G, the left inequality and the equality come respectively from Hiraguchi's and Dilworth's theorem, and dim, a! denote respectively the usual order dimension and stability number.) 1.4. Some easy cases a(G) = 0 e p(G) = 1= dim(G') e G' is a total order. a(G) = 1 3 p(G) = dim(G') = 2. Furthermore when p(G) = 2, we know the exact value of the jump number (algorithmically) and we have a characterization of graphs such that p(G) = 2 and a ( G ) = k. (It could be a good exercise for a reader who wants to get himself familiar with the jump number.) On the other hand, if p(G) denotes the length of a maximum path in G, we trivially have: P(G) a d G ) .
(10)
Furthermore for connected graphs: V T E ~ ( G )U(T, , G) = a ( G ) iff Gh is a complete multipartite graph.
M. Chein, M . Habib
192
1.5. The bipartite case
Let G be a bipartite dag, then for any T E ~ ( Gthe ) , steps of G constitute a matching of G without alternating cycle. Besides, the converse is also true, to any matching M without alternating cycle, we can associate T E ~ ( G such ) that the steps of T are the edges of M. For any bipartite graph G:
1 M matching without alternating cycle}.
k ( G )= max{(M)
(1 1)
These particular matchings also appear in some matroidal problems, Krogdahl in [9] called them clean matchings. Let B be a basis of a matroid .AX over a set E, and let DPG(B) be the biparite graph associated with the dependence relation in A as follows: DPG(B)= (B, E - B,F)such that for every x E E - B,its neighbours in B are the elements of the unique circuit of B U{x}. Then for every clean matching M of DPG(B), B'= M ( B )= B - (Bfl M ) U ( ( E- B)nM) is also a basis of A. Nevertheless, if we consider a bipartite graph G = (X, Y, F), then it is wellknown: 8 (G) = { Z E X U Y 1 Z = M(X), M matching of G}
constitutes the basis set of a matroid over X U Y. But
I
8 ' ( G )= { Z E XU Y Z = M ( X ) , M clean matching of G}
does not constitute the basis set of some matroid as the following example shows:
B1= b4, XS, yl. y2, y3} = MAX) with Ml = {xlyl, x2y2, x3yd and
B2= b1, y2, y3, Y,, ys1= M 2 ( X ) with M 2 =b2y2, x3y3, x4y4,xSys}. Still B1 and B2 do not satisfy the exchange axiom. If for a bipartite graph G, 8 ' ( G ) is a matroid, then the jump number problem is equivalent in this matroid to the search for the most distant basis (with respect to the symmetric difference) of a given basis. Two problems related to the bipartite case are given in the problem list (same issue).
193
The jump number of dags and posets
1.6. Some other results
Chaty, Chein, Cogis, Habib, Martin, and Petolla have obtained upper and lower bounds for the jump number, decomposition theorems, polynomial algorithms for trees, (O(nZ)), series-parallel dags (O(n2)),unicyclic (O(n3))(see (3-81). Independently, Rival has worked on the jump number problem and he has constructed important tools that lead, among other results, to polynomial algorithm for posets without particular cycles [lo]. Finally, as far as we know, the jump number problem is still unclassified as to its complexity. 1.7. Generalizations
The previous definitions of the jump number lead to several generalizations. Among them one especially is noteworthy. Adding numerical valuation to the arcs we obtain a travelling salesman problem with precedence constraints. We describe this now. A slight modification of an example given by Bellman et al., called a mathematician’s holiday in [2]: As quickly as possible execute the following tasks: a -drop letters in a mail box. b -buy stamps. c -buy some bread d-get some cash at the bank. e -play tennis. f - fill up the car tank. g -go swimming.
bu
a
9 e
f
XO
One can formulate this problem as follows: find a hamiltonian circuit of minimum cost xoxl . * x,x0 such that if xy E U, then x = 4, y = xi and i <j (where xo is the starting point and U the arc set of the precedence constraint graph G). In fact, the jump number is a particular case of the above problem. Since for a given dag G, we just have to add a unique sink xo (if G has more than one sink) and then take the following 0-1 valuation u: u(xy) = 0 iff xy is an arc of G‘.
References [l] M. Barbut, Note sur les ordres totaux ? distance i minimum d’une relation binaire donnee, Math. Sci. Humaines 17 (1966) 47-49. [2] R. Bellman, K.L. Cooke and J.A. Lockett, Algorithms, Graphs and Computers (Academic Ress, New York, 1970). [3] G. Chaty, M. Chein, P. Martin and G. Petolla, Some results about the number of jumps in acircuit digraphs, Utilitas Math., Cong. Num. X (1974) 267-279.
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M. Chein,M. Habib
[4] G. Chaty and M. Chein, Ordered matchings and matchings without alternating cycles in bipartite graphs, Utilitas Math. 16 (1979) 183-187. [5] M. Chein and P. Martin, Sur le nombre de sauts d'une for&t,C.R. Acad. Sci. Paris 275 (1972) A 159-1 61. [6] 0. Cogis and M. Habib, Nombre de saws et graphes s6rie-parallbles. R.A.I.R.O. Inf. Theor. 13 (1) (1979) 3-18. [7] M. Habib and G. Lambert, Some results about the minimal cardinal of a path set carrying out a partition of the vertices and the jump number of acircuit digraphs, Utilitas Math., Cong. Num. XIV (1975) 367-384. [8] M. Habib, Partitions en chemins des sommets et sauts dans les graphes sans circuit, These 3Cme cycle, UniversitC P. et M. Curie (1975). [9] S. Krogdahl, the dependence graph for basis in matroids, Discrete Math. 19 (1977) 47-59. [lo] I. Rival, private communication. [ll] E. Szpilrajn, Sur l'extension d'un ordre partiel, Fund. Math. 16 (1930) 386-389.
Annals of Discrete Mathematics 9 (1980)195-197 @ North-Holland Publishing Company
THE PATH-NUMBERS OF SOME MULTIPARTITE GRAPHS Bernard PEROCHE I. U.T. Villebaneuse, Unioersite' Pan's XIII,
rue J.B. Clkrnenl, 93430 Villetaneuse, France
1. Introduction and definitions A simple graph is a couple G = (X, E) such that E c 9 , ( X ) , the set of subsets of X of cardinality 2; X is the set of vertices of G and E the set of edges of G. From now on, we shall say graph for simple graph. An elementary path of G is a sequence of vertices [xl, x2, . . . , x p ] with x i # xi when i f j and such that {xi, E for l = z i s p - 1 . In the following, we shall say path for elementary path. The definitions not given here can be found in [2]. Our notations are as follows: K,, denote the complete graph on n vertices, K,,,, denote the balanced complete r-partite graph K(n, n, . . . , n), P, denote a path of length n and C,,an elementary cycle of length n. Let G be a graph and 9 a family of paths of G. If each edge of G lies on at least one element of 9, we shall say that 8 is a path-covering of G ; if each edge of G lies on exactly one element of 9,we shall say that P is a path-partition of G. The path-number of G, .rr(G)-(resp. the unrestricted path-number of G, p ( G ) )is the minimum cardinality of a path-partition (resp. a path-covering). Let H be a graph having r vertices xl, x2, . . . ,q.Let 9 ' be a family of r edgeless O), where Vi, lXil= n, n 3 1 and where Xin4. = O if i # j . We graphs S l = (y, denote H(S;, S&-. . . , S:) = (X, E) the graph obtained by substituting the graphs S l for the vertices xi and defined by:
x = u xi. I
{x, y} E E iff x E Xi, y E 3.with i # j and { x i , xi} is an edge of H.
In this paper, we give the value of .rr(H(S?,.. . , S:)) when H is a path of length (r-1), a cycle of length r or the complete graph K,. In the last case, K,(S;, . . . , S:) is K,,,,, so we obtain a generalisation of some results of [7] and [8]. We also deduce from these results the value of p ( H ( S ; , . . . , SF)), unless when H = K, with r even and n odd. 195
B. Pkroche
196
2. Known results If G = (X, E), we denote A(G) the maximum degree of G,
i(G) the number of odd vertices of G. Then, the following results will be needed:
Property 1 (see [3]). (i) If
then k ( G ) sw ( G ) (ii)
Theorem 2 (see [5]). I f i ( G )= 1x1, then .rr(G) = f 1x1. Theorem 3 (see [l]). If n is odd and r even, K,,, is the union of & n ( r - l ) - l edge-disjoint Hamiltonian cycles and a 1-factor. In the other cases, K,,, is the union of $n(r- 1) edge-disjoint Hamiltonian cycles.
3. Main results
Property 1. .rr(Pr-l(S:,. . . , S:)) = n for r = 3. Sketch of the proof. As k(G) = n, we prove the property in constructing n paths partitioning the edges of the graph. Corollary 1. p(P,-,(S;, . . . , S:)) = n for r 3 3.
Proof. We apply Property l(ii) of Section 2.
Corollary 2. For any graph H having r vertices and such that .rr(H)= k, .rr(H(S:, . . . , S:)) s kn. Corollary 3. For any graph H having r vertices, for any family of r graphs Gi = (Xi,Ei)with Vi, [Xi[= n, we have, for an obvious definition of H ( G 1 , .. . , G,): .rr(H(G,, . . . , G , ) ) s C .rr(Gi)+n.rr(H). 1
The path-numbers of some multipartite graphs
197
Property 2. 7r(C,(S;I,. . . , S:)) = n + 1 for r 3 3. Proof. The proof is also constructive, and uses the paths constructed in the proof of Property 1.
Corollary. p(C,(S; ,..., S:))=n+l for r s 3 . We now study the case H = K , (for r a 2 ) :
Lemma 1. r ( K z k x 2=) 2k. Lemma 2. r ( K ( 2 k + 1 ) x=22k ) +1 The proofs of these two lemmas are constructive. By using Theorem 2 of Section 2, results of [l], Property l(ii) of Section 2 , Lemmas 1 and 2 of Section 3 and constructive proofs, we can show:
Theorem 3. 7 r ( K r x , ) = p ( K r x , ) = ~ n ( r - l ) + lunless , r is even and n odd. In this last case, there is a correcting term: (in-1) for And we:
Conjecture. There is a correcting term:
(-4)
T.
for p in the last case.
References [l] B. Auerbach and R. Laskar, On decomposition of r-partite graphs into edge-disjoint hamiltonian circuits, Discrete Math. 14 (1976) 265-268. [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [3] G. Chaty and M. Chein, Path-numbers of k-graphs and symmetric digraphs, Proc. 7th Southeastern Conference on Graph Theory, Computing and Combinatory (1976) 203-216. [4] F. Harary, Covering and packing in graphs I, Ann. N.Y. Acad. Sci. 175 (1970) 198-205 [5] L. Lovasz, On covering of graphs, in: P. Erdos and G. Katona, Eds., Theory of Graphs (Academic Press, New York, 1968) 231-236. [6] B. Ptroche, The path-numbers of some multipartite graphs, Rapport de Recherche No. 14 du G.R. 22 du C.N.R.S., Paris VI (1979). [7] R.G. Stanton, D.D. Cowan and L.O. James, Some results on path numbers, Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing (1970) 112-135. [8] R.G. Stanton, D.D. Cowan and L.O. James, Tripartite path numbers, in: R.C. Read, Ed., Graph Theory and Computing (Academic Press, New York, 1972) 285-294.
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Annals of Discrete Mathematics 9 (1980)199-204 @ North-Holland Publishing Company
A CONSTRUCTION METHOD FOR MINIMALLY k-EDGE-CONNECTED GRAPHS M. HABIB Institur de Programmafion, Unioersiti Paris VI, 4, place Jussieu, 75230, Paris Cedex 5, France
B. PEROCHE I . U.T. Villetaneuse, Unioersiti Paris-Nord, rue J.B. Clirnent, 93430, Villetaneuse, France We give here a recursive construction method for minimally k-edge-connected graphs, inspired from Mader’s work. This method uses two operations on graphs and starts with a graph on two vertices when k is even, or with the class of the quasi-regular k-edge-connected graphs otherwise. Happily this last class is well-known as Kotzig gave a recursive construction method using only one operation.
Rhsumh: Nous prtsentons une mtthode de construction rtcursive des graphes k-ar2teconnexes minimaux, inspirte des travaux de Mader, permettant d’obtenir ces graphes B I’aide de 2 opkrations i3 partir du graphe a deux sommets pour k pair et de la classe des graphes quasi-rtguliers pour k impair (cette dernikre classe ayant e t t caracttriste rtcursivement par Kotzig, a I’aide d’une seule optration).
1. Introduction We deal here with finite undirected loopless graphs (multiple edges are al, ( G ) )be such a graph, where V ( G )and E ( G ) denote lowed). Let G = ( V ( G ) E respectively the vertices and edges sets of G. For every x # y, x, y E V ( G ) ,we denote by [x, y]G the set of all edges joining x and y. Also we denote by N ( x )={y E V ( G ) :[x, Y]G#@} the set of all neighborn of x. Furthermore A ( x , y, G ) for every x # y, x, y E V ( G ) , will be the maximum number of edge-disjoint paths joining x and y. We also define the edgeconnectivity of G to be A(G)= minx,ysv~~~{A(x, y, G ) }(other classical notations of graph theory used here can be found in Berge [l]). Let qk be the class of all minimally k-edge-connected graphs (i.e. graphs such that h(G)= k and for every edge e, A(G - e) < k). We need also the definition of Ze,the subclass of q k made with the quasi-regular graphs (i.e. graphs the vertices of which have degree k except at most one of degree k 1). About edge-connectivity there exist some well-known operations on graphs, and we shall now describe three of them using Mader’s notations. 0,:choose rn different edges el, . . . ,em of E ( G ) , subdivide ei E [x, y]G by a
+
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M.Habib and B. Peroche
200
vertex xi$ V ( G ) (i.e. replace the edge ei by the edges e ; E [ x ,x i l G and eyE [xi, yl0), and identify xi,. . . ,x, to a vertex z $ V ( G ) . 0;: proceed as in Om,then choose a vertex y E V ( G )and add a new edge joining z and y. 0;: proceed as in Om,thereby constructing the new graph G ' , choose again rn different edges of E ( G ' ) : f l , . . . ,f m not all incident to z, subdivide f i by a vertex x { $ V(G'),identify x i , . . . , x), to a vertex z ' $ V(G')and add a new edge joining z and z'. These operations yield the following results:
Theorem 1.1 (Kotzig [4]). The graphs of %$ arise by repeated application of (respectively OfklzJ) when k is even (resp. o d d ) from the graph K,k. Where K : is the only graph of
%k
Ok/2
on two vertices (i.e. K i has k parallel edges).
Theorem 1.2 (Mader [7]). Starting with K:k we obtain all graphs whose edgeconnectivity is greater or equal to 2k, by successive addition of edges and repeated application of 0,. Theorem 1.3 (Mader [7]). Starting from K i k + l or K:'+' we obtain all graphs whose edge-connectivity is greater or equal to 2k + 1 by successive addition of edges and repeated application of 0: and 0;. Where Kik+' is the graph on three vertices, say a, b, c, with respectively k + 1, k + 1, k parallel edges joining respectively a and b, b and c, c and a. Unfortunately, the whole class %k can't be obtained by these operations if we ask for all the considered graphs to be in ( e k . So we shall introduce two new operations on graphs that enable us to define %k by repeated application of these operations starting from K," when k is even, or starting from %$ when k is odd. Besides Chaty and Chein [2] have recently given a characterization of (e2, that leads in this case to a recursive construction which is different from the one proposed here.
2. Operations ae and
Pe
For A c V ( G ) ,A#P, we denote o ( A )={e E E ( G ) :e has exactly one end in A}. If lw(A)I= k, then w ( A ) is called a k-cut of G and when \ A [ #1 and / A [ #IV(G)l- 1, w ( A ) is called a proper cut. Besides, we call removable any edge e E E ( G ) such that h(G-e)=h(G) and we denote Ek(G)= {e E [x, y I G : min{d(x), d(y))> k).
Minimally k-edge-connected graphs
201
Let us give first a few elementary remarks which are obvious: (1) If G E %k, then V x E V(G), d ( x ) > k. (2) If h(G) = k, any removable edge is contained in Ek(G). (3) If G E %k, then for every e E E(G) (respectively e E Ek(G)), there exists at least one k-cut (resp. one proper k-cut) containing e. Now we shall define two operations which yield a construction of %k. (1) Operation ( Y k : Let G,, G2 be two graphs such that V(G,) n V( G2)= 8, and with x1 E V(Gl), x2 E V(G2) such that dG,(xl)= dG2(x,)= k. If a graph G admits a proper k-cut w(A) such that the graphs GA and GA respectively obtained by contracting in G, the subset A in a single vertex a (resp. in 6 ) are respectively isomorphic to G1, G2 with a associated to x1 (resp. ii to x 2 ) , then we denote G = ak(G1, G2).
Property 2.1. If G,, G, E %,and G = ak(G1,G2), then G E % k . The proof is direct when using the Lemma 2 of Mader’s work in [7].
Property 2.2. Let G E ( e k , with Ek(G)#8, then there exist G1,G2e%k such that G = %(GI, G2). Immediate. ( 2 ) Operation &: Let G, G‘ be two graphs with IV(G’)l= I V(G)I+ 1, we denote G’ = &[F, A](G) if there exist F c E(G) and A c V(G) such that: - E k ( G ) c F and IFIS LbkJ, - A = { a l , . . . , a,} is an independent set of G-F, and every vertex x adjacent to A must verify d G ( x )= k, and if G’ is isomorphic to the graph G ’ obtained from G by the following construction. Subdivide each edge ei of F by a vertex x i $ V(G) and identify { x l , . . . , x,,,} to a unique vertex z $ V(G), create yi parallel edges joining z and ai, for every i = 1,.. . , p , in order to obtain dG,,(Z)= k.
Property 2.3. Let G E %k and G‘ = &[F, A](G), then G‘ E % k . Proof. With the above definition we have d G , ( z ) = k (for brevity we keep the same notations in GI’ and G’). Furthermore, V x E V(G)-A: d G t ( x )= d G ( x ) . Similarly V i = 1, . . . , p, d G e ( a i=) d G ( a i + ) yi. But we can use the following obvious property: If G admits a vertex z such that d ( z )2 k and V x , y E V( G) - z, h(x, y, G) 3 k, then h (G)5 k. So applying this property we have: h(G’) 3 k and as d G . ( z )= k, then h(G’) = k. Furthermore, E , (G’) = (4 by construction and with Remark 2, G’E %k.
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M.Habib and B. Peroche
3. The construction method for qk Definitions. Let G be a graph and z E V(G). If there exist two edges e E [z, Y]G. e ’ E [ z , y’]G, y # y‘, then if f$E(G) we call Gee‘the following graph: V(Gee’)= V(G) and E(G“’)=(E(G)-{e, e ‘ } ) U { f } ,where y and y’ are the ends of f . Furthermore, when for every x , x’ E V(G)- z, x f x ’ : A ( x , x ‘ , P ’ ) = A ( x , x’, G), then we say that G has an admissible lifting {e, e’} at 2. Property 3.1. Let G be a graph wirh IV(G)133 and a vertex z such that d ( z )= 2 , then G has an admissible lifting at z. Immediate. On this concept Mader in [7], first proved the very strong following result, which was conjectured in a weaker form by Lovasz in [5], and proved in the particular case of eulerian graphs in [ 6 ] .
Theorem 3.1 (Mader). Let z be a non-separating vertex of G, such that d ( z ) a 4 and lN(z)la2, then G has an admissible lifting in z. We have now to distinguish between two different cases following the parity of k. (A) k is euen.
Theorem 2. Let G E %k wirh 1 V(G)I 3 3 and Ek(G)= (a, then it exists G’E%k such that G = &(G’).
Proof. Let z be any vertex of G with d ( z )= k (it is well-known that every graph of %k has at least two of them). Necessarily, z is not a separating vertex of G. (1) Let us suppose (N(z)l= 1, i.e. z has a unique incident vertex, say x, then, x is necessarily a separating vertex of G and so d ( x )3 2 - k. Furthermore E,,(G)= (a implies: Vy€N(x), d(y)= k, and as trivially G - z E % ~ , thus G = Pk[g, {x}l(G-z)* (2) IN(z)I 3 2, then using Mader’s theorem, or with the Property 3.1 in the case where k = 2, there exist a sequence of graphs G = Go,GI, . . . , Gh such that: Gi+,= GI“’ and {e, e’} is an admissible lifting of Gi at z, moreover Gh has no admissible lifting at z. We notice that A(G, - z ) a k (by the definition of admissible liftings), besides the degree of the vertices of V(G)- z are not modified by these liftings SO with the previous remark there exist at least one other vertex 2’ with dG(Z‘)= dGh(Z’)= k. Hence A (G - z) = k. But we may have G,, - z $ %&,because of removable edges. So let us denote F‘ the set of edges such that Gh- z - F’ E %k. Let us denote H = G - F’. As Ek(G)= (a, all the edges of F‘ come from the
Minimally k-edge -connected graphs
203
previous liftings. Like Gh, H has no admissible lifting in z ; but this can only happen in a few cases. (a) IN(z)l> 1, but d , ( z ) < 4 . Here we have only two possibilities left as d H ( z ) must be even. - d H ( Z ) = O , then G = & [ F , A ] ( H - z ) as H-zE%k and where F = L - F ’ , L = E ( H ) - E ( G ) ; A ={q: it exists e’EF’ incident to q}; ?/i = ( { e ’ ~ F e’nq#P)}I ’l V a E A, d G ( a ) > k as there was a removable edge incident to a in Gh, thus every edge e E [ a , a’],-, with a, a ’ € A must be in F, (if not Ek(G)= d! is impossible), hence A is an independent set in N- z - F, furthermore with the same reason V y E N(A), we have dH-z(y) = k. So all conditions needed by the operation & are satisfied. - d H ( z ) = 2 , but then there is a contradiction, as by Property 3.1, Gh has an admissible lifting in z. (b) N ( z )= {x} and I[x, zIHI = h. Then for the same reason as above we have G=pk[F,A](H-z) where F = L - F ‘ , L = E ( H ) - E ( G ) , A = { a i J 3 e ’ ~ Finci’ dent to a i }U x, yi = I{e’ E F’: e’ n q # 811, ?/* = h.
Theorem 3.3. The graphs of P k from the graph Ki.
%k
arise by repeated application of operations
ffk
and
Proof. Let G E %k, if Ek(G)= P, the result falls from the previous theorem, when E,(G)# P, by Property 2.2 there exists a family of graphs G I , .. . , G,, such that V i Gi E %k, and G can be obtained from this family only with uses of f f k .
Remark. As an immediate corollary we may obtain from this theorem, the Kotzig’s Theorem 1.1. (B) k is odd. Let us assume k
=2p
+ 1.
Lemma. Let G E %k - x k with no proper k-cut. Then for every e E [x, y]G such that d ( x ) = k and d ( y ) > k and for every z, t E V(G)-x, we have A(z, t, G - e ) > k. The proof is straightforward. Furthermore, we have a theorem analogous with Theorem 3.2.
Theorem 3.4. Let G E %k - Xkwith IV(G)I5 3 and without any proper k-cut. Then it exists G’E %k such that G = &(GI). Proof. The proof also is analogous with the one we have given for Theorem 3.2. As G E %k - xk, there exist at least one vertex z E V(G) and one edge e E [z, y]G such that d ( z )= k and d ( y ) > k. So we can apply the above lemma, and if we consider the graph G‘ = G- e, then d G , ( z )= 2 p and thus we can follow exactly the proof given in Theorem 3.2.
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M.Habib and B. Peroche
We can now formulate:
Theorem 3.5. The graphs of operations ak, P k .
%k
- %& arise from
x k
by repeated application of
4. Application We can deduce very easily from the previous work and upper bound for IE(G)I when G E %k.
Property 4.1. Let G E % ~ with E k ( G ) = @ ,then = I V(G)I).
I E ( G ) I S k ( n - l ) (where
Proof. We define S, ={x E V ( G ) :d ( x )= k } , then naturally V ( G ) -S, is an independent set of G. So IE(G)I= Iw(sk)l+IE(G(S,))I where G ( S k )denote the subgraph Of G induced by s k . But Iw(sk)l=k I s k l - 2 lE(G(sk))l,then it follows: IE(G)I= k lSk\-\E(G(Sk))l, the maximum of this expression is obtained when = n - 1 and S k is an independent set, and we have lE(G)IS k ( n - 1). This give us a result due to Dalmazzo in [3].
Property 4.2. Let G E% k , then IE(G)lG k ( n - 1). Immediate, when applying the previous Property 4.1 and the Property 2.2.
References [11 C. Berge, Graphs and Hypergraphs (North-Hlland, Amsterdam, 1973). [2] G. Chaty and M. Chein, 2 minimally edge connected graphs, J. Graph Theory 3 (1979) 15-22. [3] M. Dalmazzo, Sur la k-connexitk et la k-forte connexitk dans les graphes, Thtse, Universiti P. et M. Curie (1978). [4] A. Kotzig, S6vilost a pravidelni s6vilost konechjlch grafov, Dissertation, Karlsuniversitat Prag, VSE Bratislava (1956). [5] L. Lovasz, in: Proceedings of the 5th British Combinatorial Conference 1975 (Utilitas Math. Publ., 1976) p. 684. [6] L. Lovasz, On some connectivity properties of eulerian graphs, Acta Math. Acad. Sci. Hungar. 28 (1976) 129-138. [7] W. Mader, A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145-1 64.
Annals of Discrete Mathematics 9 (1980)205-216 @ North-Holland Publishing Company
THE AUTOMORPHISM GROUP OF A MATROID Annie ASTIE-VIDAL UER de Marhirnatiques, Universiri Paul Sabafier, 31077-Toulouse Cedex, France Ce papier est une synthttse des travaux de difftrents auteurs concernant le groupe des automorphismes d’un matro’ide. On peut distinguer trois parties: La premibre partie est consacr6e au problbme, qu’il est classique d’ttudier pour difftrentes structures (graphes, plans projectifs, . . .): Etant donnt un groupe, existe-t-il un matro’ide dont le groupe d’automorphismes soit isomorphe ii ce groupe. Si la rtponse est affirmative, chercher ti rtsoudre le meme problttme pour une classe particulikre de matro‘ides (ex: graphiques, connexe, de rang donn6, etc.). Nous ne donnerons ici que I’bnonct des rtsultats obtenus par difftrents auteurs. La deuxikme partie est I’ttude, fake par l’auteur, des matro’ides symttriques. On y donne quelques rtsultats gtntraux sur les matro‘ides posskdant un groupe d’automorphismes transitif sur les Cltments, ou transitif sur les bases. On y determine entibrement les rnatro‘ides vkrifiant la condition suivante: Si 4 bases B,, Bi, B,, B, sont telles que IBi n B i l = (B, nB,l, alors il existe un automorphismeo du matro’ide tel que o ( B i )= B, et o ( B i )= B,(la preuve utilise des rksultats de S. Maurer). La troisittme partie est tgalement due ii l’auteur. Elle traite du groupe d’automorphismes du graphe des bases d’un matro’ide. Le groupe d’automorphismes du matroiile peut-&re considCrC cornme un sous-groupe de ce groupe. Nous avons d’abord ttabli une C.N.S. pour qu’il y ait Bgalitt entre ces deux groupes; puis, pour rtpondre une question du Professeur M.P. Schutzenberger, nous avons dttermint la structure du groupe quotient de ces deux groupes, quand ce quotient existe.
0. Introduction Let M be a matroid on a finite set E. An automorphism of M may be defined in many different ways. For instance we can say that u is an automorphism of M if it is a permutation of E satisfying one of the following equivalent conditions: (i) X is an independent set of M iff u ( X ) is an independent set of M. (ii) B is a base of M iff u(B)is a base of M. (iii) C is a circuit of M iff a ( C ) is a circuit of M. etc. Let A(M) denote the automorphism group of M. Let M* denote the dual matroid of M, that is the matroid on the same set E having as bases the complementary sets of the bases of M. Clearly M and M* have the same automorphism group (A(M) = A(M*)). We say that the matroid M is self-dual if it is isomorphic to its dual , ( M - M * ) , and that it is identically self-dual if it is its own dual (M= M*).
Examples of graphic matroids. Let M be the cycle matroid of a graph G. Any automorphism of G induces an automorphism of M but M can have automorphisms not obtained from automorphisms of G and, in general, A(M) # A(G). For 205
206
A . Astid-Vidal
example, if G is the n-cycle, A ( G )is the dihedral group of order 2n, but A ( M ) is the symmetric group of degree n (and order n ! ) . From a result of Whitney [17], we know that if G is 3-connected without loops, thus A ( M )= A(G). The above example shows that, if M is the cycle matroid of a 3-connected graph without loop G, the problem of determining the automorphism group of M is the same problem as determining the automorphism group of the graph G ; but, generally, a matroid is not the cycle matroid of a graph.
1. Matroids with given group and given matroid theoretical properties Given a finite group H, Frucht [7] proved that there exists a finite graph G such that the automorphism group of G is isomorphic to H, and Sabidussi [15] proved that there exists a finite k-connected graph with an automorphism group isomorphic to H, for any k. These two results combined with the result of Whitney given in the preceding example of graphic matroids prove easily the following:
Theorem la. Given a finite group H, there exists a matroid M with automorphism group isomorphic to H. In fact, the proof of Theorem l a allows us to give a stronger result:
Theorem lb. Given a finite group H, there exists a finite, connected, graphic matroid with automorphism group isomorphic to H. Piff [14] proved similar results for other classes of matroids:
Theorem 2 (Piff). For any finite group H, (i) there exists a graphic, non transversal, connected matroid M with A ( M ) = H. (ii) there exists a non-binary, transversal matroid with A ( M )= H. Babai [3] studied vector-representable matroids of given rank with given automorphism group (matroids and groups may be infinite).
Theorem 3 (Babai). Given a group H and an integer k 2 3 , there exists a matroid M, of rank k, such that: (i) A ( M ) = H ; (ii) any (k- 1)-set is independent in M ; (iii) M is representable over any field F of power lFl3lHl for infinite H and IF1 3 f (\HI, k) f or finite H (where f ( n , k) takes finite values).
207
The automorphism group of a matroid
Theorem 4 (Babai). Given k 2 3, a set S and a function g :2” +Groups, there is a matroid M = M ( E ) such that: (i) E =I S (ii) r k ( M ) = rk(M\S) = k. (iii) A ( M \ T ) = g ( T ) ; VTGS. (iv) M is representable over any sufficiently large field. 2. Symmetric matroids This paragraph is the personal contribution of the author to the study of symmetric matroids. Here we consider matroids on a finite set E, and defined by their set of bases 9. We denote M = (E, a),n denotes the cardinality of E and m the cardinality of 9,9 ={I?,;i = 1 , 2 , . . . , m}. S,, denotes the symmetric group on E and S,,,the one on 9. Using these notations, an automorphism a of M can be defined as a permutation on E inducing a permutation on 9, that is: u E A ( M ) ( S a E S,,
and 315E S, :a ( B i )= BG(il
Thus, A ( M ) induces a subgroup of S,, this subgroup will be denoted by A ( M ) . The following lemma is easy to prove (see [ 2 ] ) .
Lemma. If M has no loops nor coloops,
A ( M ) is isomorphic
to A ( M ) .
Remark. If L is the set of loops and L* the set of coloops of M, and M’= (M/E-LUL”), that is the matroid obtained from M by deleting loops and coloops, we have: A ( M ) - SL@SL*@A(M’), direct product of the symmetric groups respectively on L and L* and the automorphism group of M’. And we can prove easily that A ( M ) = A ( M ’ ) . From now we always consider matroids without loops nor coloops. The preceding remark shows that this does not restrict the generality of the problem.
Definitions. (1) The matroid M is element-symmetric if the permutation group A ( M ) acts transitively on E, that is: Vx, y
E E ;3
a A~( M ) : a ( x ) = y .
(2) M is base-symmetric if the permutation group A ( M ) acts transitively on 9, that is:
VBi,BiE 9, 3a E A ( M ) : u ( B i )= Bi.
A. Astit-Vidal
208
(3) M is strongly base-symmetric if it satisfies:
VBi,Bi,Bk,B,E!i%:IBiflBjI=IBknBII; then 3a E A ( M ) such that: u(Bi)= Bk,a(Bi)= B,. Another way of defining strong base-symmetry is to consider the action of the permutation group A(M) on the Cartesian product 9 X 9. Denote Ak the subsets of 9 X 9 defined by:
Ak={(Bi,Bi): IBi-BiI=k}={(Bi,Bi):lBinBiI=h-k} for k = 0,1, . . . , 6o with 6,
= sup{lBi - BiI; Bi, Bi E 93)
( 36 , s h).
We can say that M is strongly base-symmetric if the orbits of the Sets A k ; k = 0, 1, . . . , 6 0 .
A ( M ) on 9 X 9 are
Using terminology of permutation groups, we know that the rank of a permutation group transitive on 9 is the number of orbits on 9 X 9. Thus a third way for defining strong base symmetry is to say that A ( M ) is a transitive group on 9 of rank 6,+1. It is clear from the definitions that strong base-symmetry implies basesymmetry. To illustrate these definitions, let’s give some examples of matroids possessing some of these symmetries.
Example 1. Clearly K : the matroid with n elements and every h-set of elements is a base has the three symmetries.
i=l
is base-symmetric but not element-symmetric except if ni = n, hi = h, Vi = 1, . . . , t ; and it is not strongly base-symmetric except if
I
V f = 1 ,. . . , t, or
ni=n, hi=l;
n i = n , hi=n-1; Vi=l, ..., t
We don’t give here the proof of this result, which is easy, we just give a particular case to illustrate why the strong base-symmetry is not true in general: Consider the matroid K f + K i (Fig. 1):
B1 ={a1 bl a2
b217
B2={a1
d217
bl
c2
%={a1 c1 a2 CZ), IB1nB21= IB, nB31= 2.
The auromorphism group of a matroid
209
Fig. 1
If
~
U
AE( M ) : (Bl,B2)+ (Bl, B3), thus
which is clearly impossible because a connected component of a matroid is necessarly mapped on a connected component.
Example 2. Let’s give an example of a matroid which is element-symmetric but not base-symmetric: G is a 3-connected graph, thus the cycle matroid M ( G ) of the graph G is connected and A[G]- A [ M ( G ) ](see the example at the beginning of this paper). Clearly the permutation group A (G) is transitive on the edges of G, thus M ( G) is an element-symmetric matroid (Fig. 2). 2
1
Fig. 2
M ( G ) is not base-symmetric because there exist non isomorphic spanning trees of G (Fig. 3).
Example 3. An example of element and base-symmetric matroid is provided by means of Jordan groups (Kantor [ll];Cameron and Deza [4]).A Jordan group G is a transitive group containing a subgroup K fixing some points and transitive on
A. Astie-Vidal
210
the set of points which are not fixed. Let H be the set of fixed points of K. { g ( H ) ;g E G } are the hyperplanes of a matroid M ;and G is a subgroup of A(M) transitive on the elements and on the bases of M. To give a more precise example (cf. [4]) we can consider E, vector space of dimension 2 over the finite field F of order 4.Let
G ={(x, y ) + (bx + c , ax + y + d ) ; a, b, c, d E F ; b f 0). Let
K = { ( x , y ) + ( b x , a x + y ) ; a, b E F , b f O } . K fixes the points of the set H = { ( O , y ) ; Y E F } and { g ( H ) ; g E G } = {(c, y + d ) ; y E F } are the hyperplanes of a matroid on E. The bases of this matroid are the couples { u, u } such that u and u do not belong to the same hyperplane. The hyperplanes are the five vertical lines and the bases are the couples { u, u} such that u and u are not on the same vertical line (Fig. 4).
Fig. 4
In Propositions 1, 2 and 3, we give some easy results concerning symmetric matroids but the main result of this paragraph is the theorem in which we determine all the strongly base-symmetric matroids.
Proposition 1. Let M be an element-symmetric matroid. The connected components {Mi;j = 1,2, . . . , t} of M are isomorphic and element-symmetric, and
A (M) = A (MI)- S, wreath-product of the symmetric group of degree t with A(M,).
Corollary.
The automorphism group of a marroid
211
Let { E i ;j = 1,.. . , t} be the partition of E corresponding to the connected components of M. Clearly, V ~ E A ( M ) , V ~ .E. .{,~t},gj’E{l,. , . . , t}:
a(Ei)=(Ei,).
Thus {Ei;j = 1 , . . . , t } is an imprimitivity system for the group A(M). From this remark, it is easy to verify that the permutations of A(M) can be written in the following form: Denote El = {(k, 1); k = 1, . . . , l } . We know that Vj = 1, . . . , t ; 3ajEA(M): ui(E,)= Ei.
The elements of Ei will be denoted: Ej={(k,j);k = l , ..., 1}
with (k,j)=ui(k,l).
With these notations:
E = {(k,j); j = 1, . . . , t ; k = 1, . . . , I } and ~ E A ( Mcan ) be written a = ( P 1 , - .* Pt; a ) & € S t , PiEA(M1) 3
and d k , j ) = (Pj(k); ( ~ ( j ) ) . The corollary follows from the fact that the automorphism group of the complete matroid K i is the symmetric group S,.
Proposition 2. A base-symmetric matroid is a sum of element and base-symmetric matroids. Corollary. A n y connected base-symmetric matroid is element-symmetric. To prove the proposition, it is sufficient to note that if A l , A,, . . . , A, denote the orbits of the permutation group A(M) on the set E, and if Mi denotes the submatroid of M induced on A, (i = 1 , . . . ,s), we have M I ; = , Mi, and Mi is clearly element and base-symmetric. The corollary follows trivially.
Proposition 3. A strongly base-symmetric matroid is element and base-symmetric. From the definition, if M is strongly base-symmetric, M is base-symmetric; and Proposition 2 implies that M = Mi with Mi element and base-symmetric. M having no coloops, we have, Vi E { 1, . . . , s}, there exists two bases B1 and B, such that IB, - B,I = 1, and B1- B2 is an element of Mi. Thus let
IB,- B,I
=1
and B1- B, = {x} element of Mi ;
IB3- B,I
=1
and B, - B, = {y} element of Mi.
and
A. AstM-Vidal
212
M being strongly base-symmetric, 3a E A(M) such that a ( B J = B , and a ( B 2 )= B,; thus a ( M i )= Mi. And we have proved that Vi, j ; 3 a € A ( M ) :a ( M i ) =Mi; thus M is elementsymmetric. The main theorem of this paragraph is the following characterization of strongly base-symmetric matroids (the proof is based on results of Maurer [13]).
Theorem. A matroid is strongly base-symmetric i f and only if it is isomorphic to one of the following:
Proof. Let L(M) denote the matroid basis graph, that is the graph with vertex-set W ; Bi and Bj being joined if and only if IB, - BjI = 1. Let N(B,,B j ) be the set of common neighbours of Bi and Bj in L ( M ) ,That is: N(Bi,B i )= {Bk E 9 : (Bi- Bk I = 1, IBi - Bk I = 1). The first step of the proof is the following lemma.
Lemma. Let M = ( E , W) be a matroid such that: (i) V B ,E 9,3Bi E W : (B,- Bi I 2 2. (ii) VB,,BjE W : IBi - Bjl = 2, implies lN(B,,Bj)l = k ( M = K : ( n s h + 2 ) , or
It is an easy result of matroid theory that, if IB, - Bjl = 2, there are only 3 possibilities for N(Bi,B j ) : IN(Bi,Bi)l = 2 , 3 or 4. Thus, we have to consider three cases: k = 2 , 3 , 4 . Here, we use results of Maurer [13] that, with our notations, may be written as (a) and (b) below: (a) If, V B i , B i € W such that I B , - B i ( = 2 , we have IN(Bi,Bj)I#3, then M = Xi K:(b) If VB,,Bj E W such that IB, - Bjl = 2, we have IN(Bi,Bj)l = 3, Then 3 8 , W ~ such that IBo - BI = 1, V B E 9. From these results we deduce that k = 3 is impossible, because if k = 3 , ( b ) implies that 3 B 0 E W such that IB, - BI = 1, V B E 9, B f B,, which contradicts our hypothesis (i). Hence k f 3, and (a) implies that
The automorphism group of a matroid
213
From this, it is easy to verify that:
k=4eM=Kk
(nah+2),
k = 2 e M = z Ki,+z K2-l. I
i
3. Structure of the automorphism group of the basis graph of a matroid Clearly the group A ( M ) ,induced by A(M) on the bases of M, is a subgroup of the automorphism group of the basis graph of M, A[L(M)]. A ( M ) being isomorphic to A(M), we simplify the notation writing that A(M) is a subgroup of A[L(M)]. The question is to determine a necessary and sufficient condition to have equality between the two groups A(M) and A[L(M)]; that is a condition so that any automorphism of the basis graph of M is induced by an automorphism of M. When we have solved this problem, Schutzenberger asked the question of determining the structure of the factor group of A[L(M)] with respect to A(M). This structure will be entirely determined in case it exists, that is in case where A(M) is an invariant subgroup of A[L(M)]. First, we give an example to exhibit how one can construct elements of “(M)IA(W.
A. Astid-Vidal
214
In fact, this example is representative of the general case and to prove our theorem, we need a result of [ 5 , 9 , 1 2 ] ,that can be formulated: If M = CieIMi denotes the canonic decomposition of M, to any element J, of A [ L ( M ) ] ,we can associate a permutation cp of the elements of M, such that V i E 1, 3j E I : cp is an isomorphism of Mi onto Mi or To determine whether such an element J,, of the form $ ( B ) = q ( B ) (cp being an isomorphism from M to M"), really does not belong to A ( M ) , we notice that:
q.
J , E A ( M ) a l I nWi)l=lIn BiI, V J r 1 (see[61) ieJ
iEJ
and here, this equivalent to
InBiI=ln BiI, V J ; isJ ieJ
which is equivalent to the existence of an isomorphism cp' from M to M* such that cp'(B,)= Bi V i E 1. We determine the matroids satisfying this condition in the following lemma (for the proof, see [ 2 ] ) .
Lemma. A matroid M
= ( E , 9 ) possesses
an isomorphism cp from M to M* such
that: cp(B)= B, V B E 9,
if and only if M =C K:. This lemma shows that we have to particularise that kind of matroids. For this, it will be useful to introduce a definition: The matroid M is strictly self-dual if M is self-dual but no component of M is isomorphic to K:. We give another useful definition: The matroid M is minimally self-dual if M is self-dual but no non-trivial component of M is self-dual. It can easily be seen that a minimally self-dual matroid must be connected or sum of two connected components (M = M I + M,,
Mi 7" M2, Mi
M)..
2 :
Theorem. Let v be the number of strictly self-dual components of M and let q be the number of minimally strictly self-dual components of M. (i) The index of the subgroup A ( M ) in A [ L ( M ) ] is: [ A [ L ( M ) ]A : ( M ) ]= 1 + v. (ii) v = 2q - 1 i f and only i f the disconnected minimally self-dual components of M are not isomorphic. Then A [ L ( M ) ] = A ( M ) - K where ; K is an abelian group such that J,, E A ( M ) , V J ,E R
The auromorphism group of a marroid
215.-
(iii) A(M) is an invariant subgroup of A [ L ( M ) ] if and only if the minimally strictly self-dual components of M are not isomorphic. Then
(direct product of q cyclic groups of order two). (iv) If the minimally strictly self-dual components of M are not isomorphic and are identically self-dual, then: A [ L ( M ) I = A ( M ) @ ( C1$ Zz)
( a matroid M is identically self-dual if M = M ) .
Corollary. Let M be a connected matroid. (i) A [ L ( M ) ]= A ( M ) , if and only i f M # M* or M - K:; (ii) A [ L ( M ) ] / A ( M ) = Z , ,if and only if M - M * and M # K : ; (iii) If M is identically self-dual and M # K:, then
A [ L ( M ) ] -A ( M ) @ Z z .
In particular S,
if n # 2 h , or n = 2 and h = l ,
A[Wc)I= S,@Sz if n = 2h and h f 1.
The corollary is a direct consequence of the theorem in the connected case. The proof of the theorem is in [ 2 ] .
References [l] A. Asti6-Vidal, Sur certains groups d’automorphismes de graphes et d’hypergraphes, ThBse, Toulouse (1976). [2] A. Astie-Vidal, Factor groups of the automorphism group of a matroid basis graph with respect to the automorphism group of the matroid, Discrete Math., to appear. [3] L. Babai, Vector representable matroids of given rank with given automorphism group, Discrete Math. 24 (1978) 119-125. [4] P.J. Cameron and M. Deza, On permutation geometries, J. London Math. SOC.,to appear. [S] W.H. Cunningham, On theorems of Berge and Fournier, in: Hypergraph Seminar, Springer Lecture Notes, 411 (Springer, Berlin, 1974) 67-74. [6] J.C. Fournier, Isomorphismes d’hypergraphes, Colloque de Bordeaux (Juin 1975). [7] R. Frucht, Herstellung von Graphen mit vorgegebener abstrakten Gruppe, Compositio Math. 6 (1938) 239-250. [8] E. Halberstadt, Sur certains sous-groupes des groupes symktriques, C.R. Acad. Sci. Paris 279 (1974) 733-735. [9] C.A. Holzman, P.G. Norton and M.D. Tobey, A graphical representation of matroids. Siam. J. Appl. Math. 25 (1973) (4).
216
A. Astit-Vidal
[lo] L.A. Kaluznin and M.H. Klin, On certain maximal subgroups of symmetric and alternating groups, Math. U.S.S.R.Sbornik 16 (1972) (1). [ l l ] W.M. Kantor, 2-transitive designs, in: M. Hall Jr. and J.H. Van Lint, eds., Combinatorics (D. Reidel, Dordrecht, 1975) 365-418. [12] S.B. Maurer, Matroid basis graphs I, J. Combin. Theory 14 (B) (1973) 216-240. [13] S.B. Maurer, Matroid basis graphs 11, J. Combin. Theory 15 (B) (1973) 121-145. [14] M.J. Piff, Some problems in combinatorial theory, Thesis, Oxford (1972). [15] G. Sabidussi, Graphs with given group and given graph-theoretical properties, Can. J. Math. 9 (1957) 515-525. [16] D.J.A. Welsh, Matroid Theory (Academic Press, New York, 1976). [17] H. Whitney, Congruent graphs and the connectivity of graphs, Am. J. Math. 54 (1932) 150-168.
Annals of Discrete Mathematics 9 (1980) 217-222 @ North-Holland Publishing Company
COMBINATORIAL ASPECTS OF CONTINUED FRACTIONS Philippe FLAJOLET IRIA, 78150 Rocquencourf France
It has been known since Euler and Gauss that certain, usually divergent, power series have (formal) continued fractions expansions of a simple form. Further examples have been subsequently given by Stieltjes and Rogers. For instance it has been found (see the classical treatises of Perron and Wall) that:
1
C n! z n =
12Z2
l-lz-
1-32-
22z2 32z2 1 - 5 ~ - -...
1 12Z2
E 2 n ~ 2=n 1-
1-
22z2
l--
32z2
...
with {E2,,}the Euler or secant numbers (C E2,(x2"/2n!) = (l/cos x ) ) . The interesting fact about these expansions, is that most divergent power series which appear there involve known combinatorial quantities; also the coefficients in the continued fractions are integers obeying simple laws. In a forthcoming paper [l], we show that almost all such continued fraction expansions can receive combinatorial proofs, from which new continued fractions are also derived. Furthermore it is known that the polynomials appearing in convergents of continued fractions satisfy certain formal orthogonality relations. Many of the classical orthogonal polynomials were first introduced in this way. The convergents of the continued fractions that can be interpreted combinatorially involve some of the most classical polynomials: the Tchebycheff, Hermite, Laguerre, Meixner and Poisson-Charlier polynomials. As a consequence, we are able to give combinatorial interpretations for the Taylor coefficients of fractions (in particular inverses) involving these classical polynomials. 217
P . Majolet
218
By-products of our treatment include a new combinatorial interpretation of the coefficients of Jacobi’s elliptic functions cn(u, k), dn(u, k ) different from Viennot’s, applications to Carlitz’s cycles of binomials coefficients and a general class of inversion relations expressible in terms of plane paths.
2.
Summary of results
Consider positive paths in the plane where steps are ascending a = I:, level c =) : 1 or descending b = steps; labelling each step in a path by its height, we have:
Theorem 1. The characteristic series of positive labelled paths has the noncommutative continued fraction expression: 1
a,, I b,
1 - C()-
1-c,-
UIV where = uw-I v.
a , I b2
1-c2--
W
I b3 ...
a2
This theorem extends previous results by Touchard [2], Jackson [3], Read [7] and the author [4]. The proof is based on elementary properties of series in non-commuting indeterminates. Combining Theorem 1 with combinatorial constructions derived from FranGon and Viennot [5] leads to Theorems 2 and 3.
Theorem 2. Let Pn,,n2,m be the number of partitions having n, singleton classes, n2 classes of cardinality 2 2 , a n d m non-singleton transient elements, then the generating function P(U,,
u2, t, 2 ) =
2
Pnl,n2.mU~~U~2tmZm+n~+2”~
nl.n2,ma0
has the expansion: 1 1u2z2
l-U,Z-
1- ( u , + 1t)z -
3U2Z2
l1--( u( 1U ]++22 * t)z-t)z -- . . .
In particular:
1B,z”= l-lz-
1 1-22--
1
S(n, k ) u k z “ =
1z2 2z2
...
1-242-
1uz2 2uz2 1- (1 u ) z --. . .
+
219
Cotnbinaforial aspects of continued fractions
(iib)
Jnzn=
1 1z2 1. 2z2 1-~ 3z2 1--. .
I
l-Z--
...
where the B, are Bell's exponential numbers; the S(n, k ) are the Stirling numbers of the 2nd kind; I, counts involutions on n; .In counts involutions on n having no fixed point, i.e. J2, = 1 * 3 * 5 * * (2n - 1);J2n+l= 0.
-
Theorem 3A. Let Pk,!,,, be the number of permutations having k minima (hence k + l maxima), 1 double rises a n d m double falls. The generating function P(u, v, w, 2) =
c Pk,l,mUkvfWmZ*k+f+m+l
has the expression: P(u, v, w,z ) =
1 1* 2uz2 1- l(v + w)z 2 . 3uZ2 1- 2(v + w)z -~. . .
In particular: (i)
,..\
1
C (n+l)! zn= nz-0
.
I -
L I .
A,+i,k+iU"Z"
(11)
=
n,kaO
(iii)
1 .2z2 2 * 3z2 1-4~-- . . .
1-22-
1 1 * 2uz2 1- 1(1+u ) z 2 . 3uz2 1-2(1+ u)--
C E ~ , + ~ Z ~ ~ + 1' =2 z 2
7
2
na0
1-
3
*
.1 -
2 . 3z2 3 *4z2
1--
...
in which A,,k is the Eulerian number counting the permutations of [l * * * n] with k rises a n d E2n+lis the odd Euler number or tangent number counting the alternating permutations of [l * * 2 n + 11.
-
220
P. Flajolet
Theorem 3B. The following expansions hold
c n!zn=
n>O
1
' 22z2
l2Z2
1-z-
1 - 3 ~ - - ...
c
1
An,kUkZn=
n, k a0
c E2,z2"
1-uz-
12uz2
1
22z2 1 - ( 1 + 2 u ) z --. . .
1
=
>
1-
l2Z2
-1--
2222
32z2 1 --. . .
1
C s(n, k ) u k z n= 1-uz-
UZ2
1 -(2+ u ) z -
(1 + u ) 2 2
...
where the An,kare the Eulerian numbers; E2, is the 2nth Euler number or secant s,,k is the Stirling number of number counting the alternating permutations of [h]; the first kind counting the permutations of [ n ] having k right-to-left minima. Other applications include permutations partitioned according to the number of their cycles of length 1 and 2, the derangements, the generalized Euler and Eulerian numbers.
Theorem 4. The coefficient c,,k in the expansion of the elliptic series
counts the alternating permutations over [2n] having k minima of even value. Theorem 4 ultimately relies (in our proof) on the addition formulae for the elliptic functions. Interpreting convergents of these fractions leads to Theorems 5 and 6.
Theorem 5A. Let BLhl be the number of partitions of [n] of width S h ; let denote the number of partitions of [ n + h ] of width h such that 1.2, . . . , h belong to different classes.
where Qh-l(z) is the hth reciprocal Charlier polynomial.
Combinatorial aspects of continued fractions
221
Theorem 5B. Let I[,“] be the number of involutions of [ n ] having width s h ; let fi:nh!h denote the number of involutions of [ n + h ] having width h and such that 1,2.. . . , h belong to different cycles of length 2. Let I$h1and IT1 be the corresponding quantities relative to involutions without fixed points. Then
where Q;-l(z) = z h H h ( l / Z ) is the hth reciprocal Hermite polynomial and Qh+l(z) = Z h H h ( ( l / Z ) - 1).
Theorem 6. Let p,”]be the number of permutations in S, having clustering
where Qh(z) is the hth reciprocal Laguerre polynomial of order 1, and N h ( z ) is the hth reciprocal Meixner-Polynomial.
3. Sequence of operations analysis of algorithms It is worth mentioning that part of this work originates in, and has applications to the analysis of the average cost of sequences of operations for the classical dynamic data structures arising in computer science. The framework has been set down by Franson and applications of continued fractions appear in joint works with Frangon and Vuillemin, e.g. [6]. It is shown there that certain linear transforms over sequences are expressible as linear integral transforms over generating functions.
References [l] P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math., to appear. [2] J. Touchard, Sur un problkme de configuration et sur les fractions continues, Can. J. Math. 4 (1952)2-25. [3] D.M. Jackson, Some results on “product-weighted lead codes”, J. Combin. Theory 25 (A) (1978) 181-187.
222
P. Flaiolet
[4] P. Flajolet Analyse d’algorithmes de manipulation de fichiers, Rapport IRIA, Rocquencourt (1978). [5] J. FranGon and G. Viennot, Permutations selon les pics, creux, doubles montees, doubles descentes, nombres d’Euler et nombres de Genocchi, Discrete Math. 28 (1979) 21-35. [6] P. Flajolet, J. Franson and J. Vuillemin, Sequence of operations analysis of dynamic data structures, 20th IEEE Symp. Foundations of Comput. Sci. (1979). [7] R.C. Read. The chord intersection problem, N.Y. Acad. Sci. 319 (1979) 444-454.
Annals of Discrete Mathematics 9 (1980) 223-224 @ North-Holland Publishing Company.
THE ASYMPTOTIC BEHAVIOR OF A CLASS OF COUNTING FUNCTIONS M. POUZET Dkpartement de Mathkmatiques, Universitk Claude Bernard, 43 boulevard du 1 1 Novembre 191 8, 69622 Villeurbanne Cedex, France
Abstract For a class C of finite relational structures (with prescribed finite signature) the profile aCof C is the function counting the number Q C ( n ) of isomorphism types of relational structures from C on n-elements set domains. We recall our result:
Theorem. If C is extensive (i.e. each element of C can be embedded in another one having a bigger domain) and closed under substructures, then: (1) aCis not decreasing, and (2) The asymptotic behavior of aCis polynomial or greater than every polynomial. Our first proof of (1) (in a similar form) occurs in [2] and, (in this form) in [4];we give a more general result in [3]. The fact that unbounded profiles are at least linear occurs in [4]and the general result is in [ 5 ] . Here we precise a fragment of (2) containing the minimal frame needed for applications in permutation groups.
Theorem. I f C is closed under substructures and has the Disjoint-EmbeddingProperty (i.e. each pair of structures of C can be embedded in another one with n+k disjoint images), then Q C ( n ) / n k is bounded iff cP,(n)s k *
(
)
Because the bound is independent of the signature, an extension of the main theorem to certain relational structures with infinite signature can be considered. In particular the main theorem for relational structures associated to a permutation group G: the asymptotic behaviour of the orbital profile of G, (Function counting the number of n-element orbits of G), is polynomial or greater than every polynomial. Finally from a characterization of structures having automorphism groups with polynomial orbital profile we obtain decidable results for their complete theories (The simplest case is the nice theorem of [l]). 223
Abstracts
224
References [l] P.J. Cameron, Transitivity of permutations groups on unordered sets, Math. Z. 148 (1976) 127-139. [2] R. Fraisse, Cours de Logique Mathkmatique, Tome 1 (Gauthier-Villars, Paris 1971). [3] M. Pouzet, Application d‘une propridte combinatoire aux groupes et aux relations, Math. Z. 150 (1976)117-134. [4]M. Pouzet, Application de la notion de relation presque enchainable au denombrement des restrictions finies d‘une relation, t~paraftre au Zeitschift fur Math. Logik 27 (1981). [5] M. Pouzet, Sur la thdorie des relations, T h b e de Doctorat d‘Etat, Lyon, Janvier 1978,No. 78.05.
THE THEOREM OF WHITNEY AND THE FOUR COLOURS CONJJXTURE RCne MALCOR Inginieur des Ponts et Chausskes, Pans
Abstract Le sous-graphe NIE des voisins d’un sommet V d’un graphe planaire saturC est considCrC. Disposons les sommets Nl sur le contour polygonal d’un disque plan & deux faces. Disposons le sommet V sur la face Nord du disque. Toutes les arCtes joignant V aux Ni sont aussi situCes sur la face Nord. Les arCtes restantes du sous graphe qui ne doivent pas couper les arktes VN,, ne peuvent Ctre placCes que sur la face Sud, et constituent par conskquent un jeu de diagonales du polygone Ni. Le sous-graphe NiE est donc homComorphe & un polygone muni d’un seul jeu de diagonales et ou montre qu’un tel graphe est trichromatique. Mais on montre aussi que le nombre chromatique de graphe des voisins d’un graphe de nombre chromatique c est au plus c-1. Si un au moins des sousgraphes de voisins est c - 1-chromatique, alors c - 1S 3, c S 4 . Le cas oh tous les graphes de voisins sont c - 2-chromatiques est ramen6 au prCcCdent, ce qui achkve la dkmonstration. It is shown that when eliminating triangles of the Cauchy second kind (triangles eliminated by removing one side), it remains one graph with an infinite face, the center of which is an Hamiltonian circuit, both for the reduced graph and for the primitive graph. This carries along the theorem of Whitney, results connected with frame theory (Maxwell and Cremona).
Annals of Discrete Mathematics 9 (1980) 225-227 @ North-Holland Publishing Company
NON-HAMILTONIAN CUBIC PLANAR GRAPHS HAVING JUST TWO TYPES OF FACES Joseph ZAKS Unioersiry of Haifa, Haifa, Israel
Let G(m, n) denote the family of all the cubic, 3-connected planar graphs having just two types of faces, m-gons and n-gons. All the members of G(3,6) were described by Grunbaum and Motzkin [4] and they are all Hamiltonian by Goodey [2] (for a description of all the members of the analogous “G(2,6)”, see [6]). Goodey [3] proved, in relation to Barnette’s Conjecture, that all the members of G(4,6) are Hamiltonian. It has been conjectured by Grunbaum and the author [6] that all the members of G(m, n) are Hamiltonian. We [7] found a non-Hamiltonian member of G(5,8); extending this result, we have established the following [8]:
Theorem 1. For every k, k > 11, G(5, k ) contains non-Hamiltonian members. Let p ( 3 ) be the shortness coefficient of the family (8 of graphs, defined (see [5]) as liminf h(G,,)/v(G,), taken over all sequences {G,}:=l of members of (8 such that v(G,,) + m as n --* m, and where h(G,) is the size of the maximum circuit in G,. Let p*(%) be similarly defined for the size of the longest paths. We have shown in [8] the following:
Theorem 2. p(G(5, k ) )< 1 holds for all k, k 3 11, and p*(G(5, k ) ) < 1 holds for all k, k b 11, except possibly for k = 14, 17, 22 and k = 5n + 5 for all n 3 2. Recently we [9] proved
Theorem 2+e. p*(G(5,9))< 1. Most of our non-Hamiltonian cubic planar graphs having just two types of faces are obtained as follows. Let B, be the block shown in Fig. 2, in which n copies of the block B1 (see Fig. 1) are stacked together. It follows easily from Lemma 2.3 of Faulkner and Younger’s [l] that: “If a planar graph G consists of two parts, B,,and H,n b 2, joined by four edges, and if a simple circuit a (or a simple path a, having none of its end points in B,) contains all the vertices of B,,then a contains exactly two of the connecting edges which do not lie on a common face”. 225
J. Zaks
226
Fig. 1.
The idea of the construction is to start with a suitable planar 3-connected graph, having only 3-valent and 4-valent vertices, and to properly replace each 4-valent vertex by a block B,, or a similar one. Fig. 3 shows how this is done in the case of k = 5 n + 3, yielding a non-Hamiltonian member of G ( 5 , 5 n + 3 ) , n 2 2. Similar constructions yield non-Hamiltonian members of G ( 5 , 5 n + t ) , for t = 4, 5, 6 and 7 and for all n 3 2. Different arrangements are applied for the cases of G(5,k ) where k = 12, 11 and 9 (for details, see [ 8 ] and [9]). The construction of members of G(5,k ) which contain no Hamiltonian paths is slightly harder; nevertheless it is possible to construct large members of G(5,k ) in which the largest path is only a fraction of the graph, for many values of k , as stated in Theorems 2 and 3 (for details, see [8] and [9]). It should be mentioned that many of our [8] examples of non-Hamiltonian graphs of G(5,k ) are, in addition, cyclically 4-connected. The following problems are open
Problem 1. Find non-Hamiltonian members of G(5, 10); probably all members of G(5,7) U G(5,6) are Hamiltonian. Problem 2. Find non-Hamiltonian members of G ( m , n), for m = 4 and n 2 7 , or m = 3 a n d 7 ~ n ~ l O ( G ( 3 , n ) = 8 fnoa rl l ) .
5-,
3
sn-I
Fig. 2.
1
Non-Hamiltonian cubic planar graphs
227
Fig. 3.
Problem 3. Is a(G(5,k)) < 1 for some k, k 3 11 or 8 s k s 9 , where w is the shortness exponent, defined (see [5])similarly to p as lim inf(log h(G,,)/log v(G,,)).
References [ l ] G.B. Faulkner and D.H. Younger, Non-Hamiltonian cubic planar maps, Discrete Math. 7 (1974) 67-74. [2] P.R.Goodey. A class of Hamiltonian polytopes, J. Graph Theory 1 (1977) 181-185. [3] P.R. Goodey. Hamiltonian circuits in polytopes with even sided faces, Israel J. Math. 22 (1975) 52-56. [4] B. Griinbaum and T.S. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Can. J. Math. 15 (1963) 744-751. [5] B. Griinbaum and H. Walther, Shortness exponents of families of graphs, J. Combin. Theory 14 (A) (1973) 364-385. [61 B. Griinbaum and J. Zaks, The existence of certain planar maps, Discrete Math. 10 (1974) 93-1 15. [7] J. Zaks, Non-Hamiltonian non-Grinbergian graphs, Discrete Math. 17 (1977) 317-321. [8] J. Zaks, Non-Hamiltonian simple 3-polytopes having just two types of faces, Discrete Math., to appear. [9] J. Zaks, Non-Hamiltonian simple planar graphs, in preparation
Added in proof. P.J. Owens solved Problem 1 by showing p”(G(5,lo))< 1 (“Shortness exponents of families of regular planar graphs with three types of faces”, typescript, submitted for publication).
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Annals of Discrete Mathematics 9 (1980) 229-233 @ North-Holland Publishing Company
CONJUNCTURALLY STABLE COALITION STRUCTURES An&& POLYMERIS Znstitut fur Operations Research, ETH Zurich, Switzerland
We consider social organizations and conflicts concerning the distribution of resources among the members of such organizations. Cooperative solutions of conflicts are defined in a manner similar to the one used in mathematical coalition games. As organizational structures manifest themselves through the coalition possibilities which they impose on the members of the organization, they influence the existence of cooperative solutions. Structures permitting solutions for all non-antagonistic conflicts are named conjuncturally stable, because they prevent any conjunctural inconsistency. Stable structures are the ones which guarantee the practicability of a simple distribution principle. They also allow purely combinatorial characterizations revealing an equivalence between stability and a further consistency property. This permits us to trace instability back to the presence of very special substructures and leads thus to a -surely simplified-structural explanation of the perseverance exhibited by social hierarchies. Many of the mathematical results for which we give new proofs are adaptations of well-known theorems of Fulkerson, Lovhz and Padberg.
A coalition structure (A, 0, B) is a triple of non-empty finite sets A, f2 and B - actors, feasible coalitions and alternative projects -such that for every pair (a, b) E A x B there is no more than one K E 0 with a E K E b. So if the entirety is conjuncturally split by an alternative project, its feasible coalitions pack the actors into disjoint subsets of A. A distribution conflict upon such a coalition structure is a pair (r, z) such that r E R, and z E RT. r represents the total amount of resources which is available for distribution when global cooperation is not "split" by an alternative project. z is the characteristic function of the conflict. It indicates the amount of resources attainable by a feasible coalition in case it participates at an alternative project. A global settlement of such a conflict will have to distribute available resources in a way discouraging alternative coalitions. Therefore we define a cooperative solution of a distribution conflict (r, z) to be a pay-off function x E R$ such that f ( A ) d r and for all K e n , 2 ( K ) Z = z ( K )(for any function X E R ? and all A ' c A let f(A'):=C{x(a): aEA'}). If for a conflict (r, z) there is an alternative project b e B such that C { z ( K ) :K E b} > r, then global cooperation unifying all actors does not appear reasonable, since more can be attained by splitting the organization into the coalitions of b. Such conflicts will be called antagonistic. Of course they do not allow any cooperative solutions, since: if x were such a solution, then
r B 2 (A)3
1 {f( K ): K E b} 5 1{z (K): K E b} > r, 229
230
A. Polymeris
a contradiction. But also non-antagonistic conflicts may turn out to be cooperatively unsolvable [4].
Distribution algorithm Given a non-antagonistic distribution conflict (r, z ) “reduce” it step by step, starting with x := 0 and repeating the following operation until z = 0: Choose an actor a E A and a pay-off amount s > 0 such that reducing r : = r - s and for all K E R with a E K, z ( K ):= max{z(K)- s, 0}, will continue to yield a nonantagonistic conflict (r, 2); then also reset x ( a ):= x ( a )+ s. This algorithm is similar to one of Fulkerson [l, p. 631. It does not always lead to the aimed end (2 = 0). It may get blocked at a dead-end, where x can not be increased any more. But if the aimed end is reached, then non-antagonism of the final pair (r, 2) clearly implies that x is a cooperative solution of the original conflict. Let us introduce the functions p and Y :RY+ R , such that for all z E RY, p(z):=max{C{z(K): a E K } : U E A and } v(z):=max(C{z(K): K E ~ }b:E B } . If z E RY and actor a E A is such that for all b E B with C { z ( K ) :K E b } = ~ ( z ) , there is a K E R with a E K E b and z ( K )>O, then we call this actor a pivot of z -since he has a certain “central position”: Any alternative project offering maximal amount of resources will need the collaboration of this actor. If (r, 2) is a non-antagonistic conflict and a E A a pivot of z, then the distribution algorithm may get started with this actor and a sufficiently small pay-off amount; since in this case v(z) and r will be reduced by the same amount, thus avoiding any antagonism. This implies:
Lemma 1. Let the distribution algorithm be applied on a conflict (r, z ) and x be the function built up by the algorithm after some iterations. If the conflict allows a cooperative solution x* with x S x * and an actor a E A wirh x ( a )< x*(a), then the algorithm may be continued choosing this actor and pay-ofl amount x*(a)- x ( a ) .
Proof. Let (r’, z’) be the partially reduced (r, 2). Further reduction may only cause troubles if v(z‘)= r‘. But then the actor a will be a pivot of z’, since if b E B is such that
r’=
{z’(K):K
E b} = v(z’),
then
r - f ( A ) > I { x * ( a ’ ) - x ( a ’ ) : a ’ E A } z = C{ i * ( K ) - P ( K ) : K €b} 2
c {z’&):KE b}
= r’
i.e., all these terms are equal, thus implying the existence of K E R with a E K E b and z ‘ ( K )= {x*(a’)- x(a’): U ’ E K}> x * ( a ) - x ( a )>o.
c
Let us now gain distance from single conflict conjunctures and focus on their structural determinants. Structures which allow cooperative solutions for all
Conjuncturally stable coalition sfructures
23 1
non-antagonistic distribution conflicts will be called conjuncturally stable. They are thus distinguished from inconsistent coalition structures, which do not settle feasibility of coalitions in a way guaranteeing (or enjoining) cooperative solvability for all “benign” conflicts and hence risk to get questioned by them. If a characteristic function z #O has no pivot, then, because of what has just been proven, the conflict ( u ( z ) ,z ) has n o cooperative solution. So the structure is conjuncturally stable if and only if every non-zero characteristic function allow pivots. Let the support of a characteristic function z be the subclass of coalitions K E 0 for which z ( K )> 0. The next result explains why, to decide stability, only the finite set of supports has to be examined.
Lemma 2. If z has minimal support among all non-zero characteristic functions which do not have pivots, then for all coalitions G, K of its support, z ( G )= z ( K ) . Proof. Let G be a coalition of the support of z which offers maximal amount z ( G ) ,t := z ( G ) ,s > O such that s s t and z* be built out of z replacing z ( G ) by s. As z does not admit pivots, u ( z * )= v(z). Apply the distribution algorithm on ( u ( z ) ,2”). It will not reduce the support of z*, since this would imply by the minimality assumption that (v(z ) , z *> is cooperatively solvable and, because of Lemma 1, z* would have a pivot a E G , and as for all b E B with C ( z ( K ) : K E b } = u ( z ) , either G E b or C { z * ( K ) :K E b } = v(z), either a E G E b or there is a K E 0 with a E K E b and z * ( K )> 0, showing that actor a would also be a pivot of z, a contradiction. Let now K be a second element of the support of z and apply the algorithm as long as one can choose pivots out of K. Let 2’ be the partially reduced z* and x ’ E R ? be the function built up by the algorithm. Then there exists a b E B with 1 {z’(K’):K’ E b } = v(z’)and G E b (since otherwise z’ would have the same pivots as the characteristic function which results out of z ’ replacing z ’ ( G )by 0). Thus u ( z ) - ( ~ - s ) S ~ { Z * ( K ’ ) - ~ ’ K’E ( K ’b)}:= u(z’)= ~ ( z ) - i ’ ( A ) i.e., ( t - s ) S i ‘ ( A ) < z ( K ) ~and t since s may be chosen arbitrary small, z ( K ) = t must hold. The formal definition of coalition structure is symmetric in A and B: To every concept or result a symmetric one may be assigned interchanging A and B. We can therefore define a property symmetric to conjunctural stability. If we assume the influence of an actor a E A at a conjuncture characterized by z to be proportional to C { z ( K ) : a E K } , we may interpret this property as a further structural consistency: It discourages “dilution” of actors searching to influence conjunctures. Its equivalence to stability is a consequence of the following self-symmetric characterization:
Theorem 3. The structure is conjuncturally stable if and only if for all characteristic functions z E (0, l}”,1 { z ( K ) :K E 0} ~ ( z v(z) ) holds.
-
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A. Polymeris
Proof. If there is a ZE{O, l}a with z # O and C { z ( K ) :K E R } > ~ ( z *) v ( z ) , then the distribution algorithm may be applied on (v(z),z) but, since the reduction preserves the above inequality, it will not reach the aimed end. Let us now assume the structure does not have the property which is symmetric to stability. Then there exists a w E{O, l}n such that, by Lemma 2, it will not allow project-pivots (defined symmetrically to actor-pivots). Hence we may choose for every b E B an a b E A such that C { w ( K ) : a , E K } = p ( w ) and for all K E R with a b E K E b , w ( K )= 0. Let z E R f such that for all K E 0, z(K):=Card{bEB: a b E K } ’ w(K). Then v(z)
-
{ z ( K ) :K E R}= Card(B) p ( w ) > v ( z ) p ( w ) . If we now apply the distribution algorithm on ( v ( z ) , z )the just determined inequality will be preserved implying, by Lemma 2, the existence of a z’E{O, l}O which does not have a pivot and C {z’(K):K E R}>v(z‘) p(z’).
-
This theorem gives a slight generalization of [2, Lovisz’s Theorem 21. It opens an efficient way to check the stability of a given structure. A coalition structure (A’, a’, B’) is a substructure of (A, R,B ) if A’ E A, 0’ ER and B ’ c B. The next result is similar to [3, Padberg’s Theorem 3.111. It allows to trace instability back to the presence of substructures with certain “circular” features :
Theorem 4. If the coalition structure (A, 0, B ) is unstable, it contains a substructure (A*,R*,B*) such that defining p* := max{Card{K E R*:a E K}: a E A}
and v* := max{Card{K E 0”: K E b}: b E B }
yields Card(A*) = Card(R*) = Card(B*) = p* * v* + 1and for all a* E A*, K* E a*, b* E B*, p*=Card{KER*: a*EK}=Card{aEA*: ~ E K * } and v* = Card{K E R*:K E b*} = Card{b E B*: K* E b}.
Proof. Let us assume z*E{O, 1)” has minimal support R* among all functions which contradict the condition of Theorem 3. If p*>O it is easy to show that (v*+ l/p*, z*) allows cooperative solutions. Hence if X * E R$ is such a solution and A*:={a€A: x*(a)>O}, p * . v*+lsCard(R*)sI{I2*(K): K E R ” } =
{Card{KE R*:a E K } * x*(a): a E A *} s p*
i*(A) s p * v* + 1
Conjuncturally stable coalition structures
233
i.e., Card(R*)=p**v * + l = p * . ?*(A), for all a*EA*, Card{KER*: a * E K } = p* and for all K*ER*, a*(K*)= 1. Because of linear algebra arguments there exists a solution x* with Card(A*)
Theorem 5. If a coalition structure is generated by an interaction graph it is conjuncturally stable if and only if the graph does not contain cycles. Proof. If the interaction graph allows a cycle, one can extract three different chains K, K’, K”ER out of it such that K fl K’ # 8, K’ r l K “ # 9,, K ” n Kf 9, and K f l K’ r lK” = 9,. Because of Theorem 3, this implies instability. Let now the graph be a forest. If the structure is unstable a substructure (A*, a*,B*) with the properties stipulated by Theorem 4 and p* > 1 should exist. Choose K’Ea*, determine H ‘ : = K ’ n A * and a minimal G’E A with a g ’ E G’ such that all chains connecting G’ to A - G’ have g‘ as element and H’c G’. If there is a K E R* with H ’ n K f 9, and g ’ # K, exchange K’ by K and repeat the computation of H’, G’ and g‘. Since the interaction graph is a forest, K and the new G‘ will be contained in the old G‘ and will not have the old g’ as element. Therefore this procedure must converge to a pair H’, g‘ such that for all K E R*, if H’r lK # 8, then g’ E K. As Card(H’) = p* and as the nodes of H’ are not all of them covered by the same coalitions of a*, there are more than p* such coalitions with g’ as element, contradicting the definition of p * . This theorem is one of the central pillars of [4] and immediately proves the main result of [5]. Social hierarchies display interaction structures which take the shape of rooted trees. The just proven result therefore leads to an explanation of their universal presence [4].
References [l] D.R. Fulkerson, Anti-blocking polyhedra, J. Combin. Theory, 12 (B) (1972). [2] L. LQVBSZ, Minimax theorems for hypergraphs, in: A. Dold and B. Eckmann, Eds., Hypergraph Seminar, Lecture Notes in Mathematics No. 41 1 (Springer. Berlin, 1974). [3] M.W. Padberg, Perfect zeroone matrices, Math. Programming 6 (1974). [4] A. Polymtris, Stabilitat gesellschaftlicher Hierarchien, eine mathematische Konflikttheorie, Diss. ETH Zurich No. 6142 (1978). [5] E.J. Cockayne, S.T. Hedetniemi and P.J. Slater, Matchings and transversals in hypergraphs, domination and independence in trees, J. Combin. Theory 26 (B) (1979).
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Annals of Discrete Mathematics 9 (1980) 235 @ North-Holland Publishing Company.
UNIT DISTANCE GRAPHS IN RATIONAL n-SPACE C. KIRANBABU Ohio State University, Columbus, OH 43210, USA
Abstract The maximum number of points in Euclidean n-space with rational coordinates, pairwise unit distance apart, is determined by methods of elementary number theory and Witts’ theorem on inner product spaces.
COUNTING THREE-CONNECTED GRAPHS T.R.S. WALSH Computing Centre, USSR Academy of Sciences, Moscow, U.S.S.R.
Abstract Labelled two-connected graphs were counted by Riddell (see F. Harary and E. Palmer, Graphical Enumeration, p. 11) and unlabelled two-connected graphs by Robinson (ibid, chapter 8), using a unique decomposition of connected graphs into two-connected components. Here, Riddell’s and Robinson’s methods are used to count labelled and unlabelled three-connected graphs, respectively, and also two-connected graphs without vertices of degree 2 (see Graph Theory Newsletter 7 , 3 , Abs 10; 7 , 6 , Abs 7; 8, 1, Abs 8; 8,3, Abs 7). In the process, a unique decomposition of 2-connected graphs into 3-connected components is deduced from B.A. Trakhtenbrot’s canonical decomposition of two-pole networks.
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Annals of Discrete Mathematics 9 (1980)237-241 @ North-Holland Publishing Company
CHARACTERIZING HYPERCUBES John D. McFALL Saint Mary's Unioersity, Department of Mathematics, Halifax, N.S. B3H3C3, Canada For each n P 1 the hypercube, Q(n), is defined to be a graph with vertices the 2" binary numbers each having n digits and two vertices are joined by an edge if and only if they differ in exactly one digit. In this note known characterizations of hypercubes as undirected graphs are surveyed and a new variation is presented.
If G is a graph we denote by VG the vertex set of G and by EG the edge set of G. All graphs considered in this paper are finite with no loops or multiple edges. If u, w E VG and are joined by a path we use d(u, w ) to represent the number of edges in a shortest path from v to w in G. The cardinality of a subset X of VG is denoted by For each positive integer n the graph Q ( n ) is defined to have vertex set the set of binary numbers each n digits in length and two vertices are joined by an edge if and only if they differ in exactly one digit. Any graph isomorphic to Q ( n ) is called an n-cube and a graph isomorphic to some Q ( n ) is called a hypercube. The first characterization of these (finite) hypercubes was given by Alvarez [l].
1x1.
Theorem 1 (Alvarez [l]). A (finite)connected graph G is a hypercube if and only if the following conditions are satisfied: (1) G is bipartite (2) there exist ul, U,E V(G) such that the diameter of G equals d(u,, u2) and for each i = 1, 2 if c, d, e, E VG are distinct and cd, ce E EG satisfy d ( ui, e ) = d ( ui, d ) = d(u,, c )+ 1, then there is a unique f E VG satisfying d(u, f ) = d(ui,c )+ 2 and fe, f d E EG. ( 3 ) If G, is a subgraph, then G2 is a subgraph of G. (4)G3 is not a subgraph of G. Graphs G,, G2, G3 are as in Figs. 1, 2 and 3 respectively. A somewhat different characterization was given by Foldes [2] in the following variants.
Theorem 2 (Foldes [2]). A finite connected graph G is a hypercube if and only if G is bipartite and any of the following equivalent conditions hold: (1) for each u, v E VG the number of shortest paths from u to v is d(u, v ) . (2) for each u, v E VG if 0s i S d(u, u ) the number of vertices z such that d ( u , z ) = i and d ( z , u ) = d ( u , u ) - i is
237
J.D. McFall
238
1
7
Fig. 1. The graph G , .
(3) for each u, v E VG the number of neighbours of u that lie on a shortest path between u and v is d ( u , v). A local characterization of hypercubes was discovered by Laborde [31.
Theorem 3 (Laborde [3]). A connected graph G is a n n-cube if and only if ( 1 ) if u, v E VG and d ( u , v ) = 2, then there are exactly two shortest paths between u and v. (2) G contains no triangles. ( 3 ) G4 is not a subgraph of G, where G, is as in Fig. 4. ( 4 ) G has 2" vertices and n 2"-' edges. Let G be a connected graph and X E VG. Define, for each k a O , L k ( X ) = { y E VG: d ( x , y ) = k } . If u, v E VG are distinct define Z(u, v ) = { y E VG: y lies on a shortest path between u and v}, and for each 0 S k S d ( u , v), &(u, v ) = { y E I(u, v): d ( u , y ) = k}. Also define w(u, v )=
max
ILk(u,v)J.
l s ksd(u,u)
For any @ # X c I ( u , v ) define m ( X )=
nxeXZ(x, v). 1
Fig. 2. The graph G,.
Characterizing hypercubes
239
Fig. 3 . The graph G,.
These concepts were introduced by the author in [4] to characterize hypercubes as in the following three theorems.
Theorem 4 (McFall [4]). Let G be a connected graph and fix x E VG. Then G is an n-cube if and only if ( 1 ) for each k 2 0 , I L k I = ( ; ) . ( 2 ) G is bipartite ( 3 ) if 0 s k s n each vertex of L k is adjacent to n - k vertices of Lk+, and to k vertices of L k (4) for 0 s k s n - 1 if u, V E Lare ~ distinct they have at most one common neighbour in Lk+L. ( 5 ) if X c L k and 1 x1= k + 1 , then the vertices in X have a common neighbour in Lk+lif and only if each pair of distinct u, v E X have a common neighbour in L k - 1 and no two diferenf pairs have the same common neighbour. Theorem 5 (McFall [4]). A connected graph G is a hypercube if and only if G is bipartite and for all distinct u, u E VG,
where [ t ] denotes the integer part of t.
Theorem 6 (McFall[4]). A connected graph G is a hypercube if and only if (1) G is bipartite.
Fig. 4. The graph G,.
J.D. McFalf
240
(2) for each distinct u, u E VG if 1G k S d ( u , u ) and y E Lk(u, u ) , then there exists a unique subset X c L,(u, u ) such that 1 x1= k, m(X)f l Lj(u,u ) = @ i f 1s j < k and m ( x ) nL k ( % u ) = { Y } . (3) for each distinct u, U E VG, if XcLl(u, u ) satisfies [XI=k, then m ( x ) nLj(u, u ) = P, if 1 c j < k and Im(x) n&(u, u)l = 1. Let G be a connected graph and let u, u E VG be distinct. A subset X of Z(u, u ) is called disjoint in I(u, u ) if for each distinct x, y EX, m({x, y } ) = { u } . Define D(u, u ) = maximum cardinality of the disjoint subsets of I(u, u ) . Using this concept we can characterize hypercubes as follows:
Theorem 7 . A connected graph G is a hypercube if and only if G is bipartite and for all distinct u, u E VG, D(u, u ) = d ( u , u ) . Proof. Suppose G is a hypercube. Then G is bipartite and if u, u E VG, then Z(u, u ) is itself a d(u, u)-cube so it suffices to take u = 1 - - * 1 and u = 0 - * * 0 in Q ( n ) and show that D(u, u ) = n. The set of neighbours of u in Q ( n ) is clearly disjoint so D(u, u)> n. Regarding the vertices of Q ( n ) as vectors in R" the condition that m({x, y } ) = { u } is equivalent to ( x , y ) = 0 (where ( x , y ) is the usual inner product in R"). Since an orthogonal set of non-zero distinct vectors in R" is linearly independent, a disjoint subset of I(u, u ) can have at most n vertices. Hence D(u, u ) ~ d ( uu ,) . Conversely we assume G is bipartite and that D(u, u ) = d ( u , u ) for all distinct u, u. It suffices to prove that (2) of Theorem 2 holds. We do this by induction on d ( u , u ) . If d(u, u ) = 1, then (2) of Theorem 2 is immediate. Let u, u E VG, d ( u , u ) > 1, and assume (2) of Theorem 2 holds for pairs of vertices whose distance is less than d ( u , u ) . Considering I(u, u ) we prove by induction that
for all 1C k =sd ( u , u ) - 1. This equality is valid for k = 1. Assume it is valid for some k =sd(u, u ) - 1. If y E Lk(w,u ) , then d ( y , u ) < d ( u , u ) and by (2) of Theorem 2 applied to I(y, u ) , y is adjacent to d ( u , u ) - k vertices in L,(y, U ) C L ~ + ~ u )(. U , But a vertex in L k + l ( W , u ) adjacent to y is in Ll(y, u ) so y is adjacent to precisely d(u, u ) - k vertices of & + I ( & IJ).Similarly each vertex in Lk+l is adjacent to k + 1 vertices of Lk(u, u ) . Hence
u)I (k + 1)= ILk(u, u)I (d(u, u ) - k )
I&ti(~,
and
Characterizing hypercubes
241
Now ILd(u.v)-I(u, u)l= ILl(u, u)l and Ld(u,u)-l(~, u ) is a disjoint subset in I(u,u ) so IL,(u, v)l< D(u, u ) = d(u, u ) . Let X be a disjoint subset in I(u, v ) with 1 x1= d ( u , u). For each x E X choose a vertex 2 EL^(^,^)-^ such that LEI(&v ) . If x, y E X are distinct, then f # 9 since m({x, y}) = { a } . Hence IL,(u, u ) l a d ( u , u). Thus condition (2) of Theorem 2 holds for I(u,v ) and the induction is complete.
References [l] L.R.Alvarez, Undirected graphs realizable as graphs of modular lattices, Can. J. Math. 17 (1965) 923-932. [2] S. Foldes, A characterization of hypercubes, Discrete Math. 17 (1977) 155-159. [3] J.M.,Laborde, Characterization locale du graphe du n-cube, Journbs combinatoires, Grenoble (June 1978). [4] J.D. McFall, Hypercubes and their characterizations, University of Waterloo, Department of Combinatorics and Optimization Research Report CORR 78-26 (August 1978).
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Annals of Discrete Mathematics 9 (1980) 243-246 @ North-Holland Publishing Company
SUR LES ORIENTATIONS ACYCLIQUES DES GEOMETRIES ORIENTEES DE RANG TROIS Raul CORDOVIL E.R. Combinatoire, U.E.R. 48, Universitk P. et M . Curie, 4 , Place Jussieu, 75230 Paris Cedex 05, France Nous dkmontrons un thtorkrne gtntralisant aux gtorn6tries orienttes un rtsultat classique de Gallai-Sylvester relatif aux plans projectifs sur un corps ordonnt. Ce thtorkme constitue une extension d’un rtsultat rtcent da B .I.Edmonds, K. Fukuda et A. Mandel et L. Lovhz. Thbreme. Soit G ( E ) une gkomktrie orient& de rang 3 , sans boucles. Soit ti, j = 2 , 3 , . . , le nombre de droites de G ( E ) constituies de j points et 2pi, i = 3 , 4 , . . . , le nombre de sousensembles A de E tels que AG est une ghnttrie acyclique comportant exactement i cocircuits positifs. Alors on a:
1. Introduction Le thCorkme suivant a CtC annoncC indkpendemment par J. Edmonds et al. [4] et L. L O V ~ S[ll] Z au cours du Colloque “Algebraic Methods in Graph Theory” (Szeged, 1978):
Theoreme 1.2. Dans toute gtomdtrie orientte (finie) de rang 3 il existe une droite contenant exactement 2 points. Ce thCorkme gCnCralise un rksultat classique de Gallai-Sylvester relatif aux plans projectifs sur un corps ordonnC. Pour I’historique de ce problkme on verra par exemple Motzkin [6]. Notre propos dans cette note est de donner une dkmonstration courte d’un rCsultat (ThCorkme 3.1) contenant le ThCorkme 1.2 en utilisant les techniques introduites par M. Las Vergnas pour l’btude du nombre d’orientations acycliques d’une gComCtrie orientCe [8,9].
2. Notations
Soit G ( E )une gComCtrie combinatoire orientCe [ l , 51 et V la collection de ses circuits signts [l]. Pour tout sous-ensemble A de E on dCsigne par AG, la 243
R. Cordouil
244
gkomktrie orientke obtenue B partir de G par changement de signes sur A, dCfinie par la collection AV de ses circuits sign&: si X = X' UX- est un circuit sign6 de Y alors AX = (AX)'U (AX)-, 0i.1 (AX)'= (X-nA) U(X+-A) et (AX)-= (X' nA) U (X-- A), est un circuit sign6 de AV (see [ I n . Une gkomktrie orientke est acyclique lorsque aucun de ses circuits signks n'est positif.
3. Problkme de GaUai-Sylvester pour les plans orient& Thh&me 3.1. Soit G ( E ) une gkomktrie orientie de rang 3, sans boucles. Soit 4, j = 2,3, . . . , le nombre de droites de G ( E )constitukes de j points et 2pi, i = 3,4, . . . , le nombre de sous-ensembles A de E tels que AG est une giomktrie acyclique comportant exactement i cocircuits positifs. Alors on a:
Lemme 3.2. (voir [3]). Soient G ( E ) une giomktrie combinatoire et t(G;5, q) le polynbme de Tutte de la gkomktrie. Alors on a:
ou T(G) est le treillis des fermis de G ( E ) et k ( Y ,X) la fonction de Mobius de T(G). Demonstrationdu 'lMo&me 3.1. D'aprBs un thkorkme de Las Vergnas [8, Proposition 8.1; 9, Theorem 3.11 on sait que C2pi = t ( G ; 2 , 0 ) 06 t ( G ; &q ) est le polynbme de Tutte de la gkomktrie G. On a donc:
Soit e un point de E. D'aprBs [9, Lemme 3.1.11 il existe 2t(G/e;2,O) orientations acycliques, obtenues de l'orientation de G ( E ) par changement de signes sur un sous-ensemble A de E, telles que e soit un point extrimal [8, 91 (ou, ce qui est kquivalent E - {e}soit une rkunion de cocircuits positifs) pour ces orientations. I1 rksulte de [ l o ]les Lemmes 1 et 2, que si G est une gkomktrie acyclique de rang 3 le nombre de points extrkmaux est kgal au nombre de cocircuits positifs. Alors on a
1ipi = C
i a3
e point de G
t(G/e;2,O) = 2
C itj.
is2
Sur les orientations acycliques des gEome‘tries orienties de rang 3
245
Donc il vient:
C (i-4)pi=2 C j4-4
ir 3
(1+
j>2
1(j--l)G),
jr 2
d’oh les CgalitCs (1) et (2).
Remarque 3.3. Camion [2, Chapter 111, ThCorbme 31, a dCmontrC que si G ( E )est un g6omCtrie coordonnable sur R (et donc orient6 [l])sans boucles et de rang r il existe au moins un sous-ensemble A de E tel que AG soit une gComCtrie acyclique comportant exactement r cocircuits positifs. Dans [9] Las Vergnas a conjecturk que ce rCsultat est encore valable pour toute g6om6trie orientCe de rang r. Notons, que d’aprbs [9, Theorem 1.31, si G est une gbomCtrie orientCe acyclique de rang r contient toujours au moins r cocircuits positifs. La conjecture est triviale pour r = 1, 2. Elle est vraie pour r = 3 d’aprb l’bgalitt (1): Corollaire 3.4. Soit G ( E )me gdomttrie onentde de rang 3, sans boucles. ll existe au moins 8 sous-ensembles A de E tels que AG est une gdomdtrie acyclique comportant exactement 3 cocircuits positifs.
Remarque 3.5. Par dCfinition un arrangement d de droites dans le plan projectif rCel P2(R) est un ensemble fini de droites non concurrentes. L‘arrangement 1 divise P2(R) en des r6gions bordCes par polygbnes convexes. Soient po = p o ( d ) , p1 = p , ( d ) , et p2 = p , ( d ) respectivement le nombre de sommets, aretes et faces d’un tel arrangement. Soient ti, j 2 2 le nombre de sommets incidents 2i exactement j droites de 1et pi, i 33 le nombre de rCgions comportant i cbtCs. Alors la relation d’Euler dans p2(R) donne po-p1+p2= 1.
(3)
I1 est clair qu’on a aussi
p2= ja2
ja2
ia3
C Pi-
(4)
ia3
I1 est connu que dans ce cas les CgalitCs (1)et (2) peuvent etre obtenues de (3) et (4)(voir [7]). Dans les conditions du ThCorbme 1 si on fait
on a bien les 6galit6s (3) et (4). Lovkz [ll] a annoncC une dkmonstration par
246
R. Cordouil
rkcurrence de la relation d’Euler p o - p1+ p 2 = 1, en donnant toutefois une interprktation 5 p , et p 2 diffkrente de la notre.
Bibliographie R. Bland et M. Las Vergnas. Orientability of matroids, J. Combin. Theory 24 (B) (1978) 94-123. P. Camion, Modules unimodulaires, J. Combin. Theory 4 (1968) 301-362. H.H. Crapo, The Tutte polynomial, Aequationes Math. 3 (1969) 21 1-229. J. Edmonds, The topology of oriented matroids, dans: Actes du Colloque “Algebraic Methods in Graph Theory”, Szeged, 1978, 9 paraitre. [ S ] J. Folkman et J. Lawrence, Oriented matroids, J. Combin. Theory 25 (1978) 199-236. [6] Th. Motzkin, The lines and planes connecting the points of a finite set, Trans. Am. Math. SOC.70 (1951) 451-464. [7] G.B. Purdy, Triangles in arrangements of lines, Discrete Math. 25 (1979) 157-163. [XI M. Las Vergnas, Matroiiies orientables, C.R. Acad. Sci. Paris, Str. A 280 (1975) 61-64. [U] M. Las Vergnas, Convexity in oriented matroids. J. Combin. Theory (B), 9 paraitre. [ 101 M. Las Vergnas, Extensions ponctuelles d’une gkomCtrie combinatoire orientke dans: Probkmes combinatoires et theorie des graphes, Actes du colloque International CNRS, No. 260, Orsay 1976 (Paris, 1978) 263-268. [ 111 L. Lovasz, Communication orale. [l] [2] [3] [4]
Annals of Discrete Mathematics 9 (1980) 247 @ North-Holland Publishing Company.
APPLICATIONS OF THE GORDAN-STIEMKE THEOREM IN COMBINATORIAL MATRIX THEORY H. SCHNEIDER University of Wisconsin, Madison, W l 53706, USA
B.D. SAUNDERS Rensselaer Polytechnic lnst., Troy, NY 12181, USA
Abstract By use of the Gordan-Stiemke Theorem of the alternative we demonstrate the similarity of four theorems in combinatorial matrix theory. Each theorem contains five equivalent conditions, one of which is the existence in a given pattern of a line-sum-symmetrix or constant-line-sum matrix which is semi-positive or strictly positive for the pattern. A generalization of the Gordon-Stiemke Theorem is stated in terms of complementary faces of the positive orthant and combinatorial applications are given. Many of our results are classical, but some are new.
NECESSARY AND SUFFICIENT CONDITIONS THAT A SUBGRAPH OF KT CAN BE PACKED IN K : E. MENDELSOHN University of Toronto, Toronto, Ontario, M5S 1A4, Canada
Abstract Necessary and sufficient conditions are given on n for each of the 14 subgraphs of KZ in order that K: may be packed.
247
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Annals of Discrete Mathematics 9 (1980)249-252 0 North-Holland Publishing Company.
THE ENUMERATION OF NONHOMEOMORPHIC GRAPHS BY EDGES Gregory SORKIN 80-06 246 Street, Bellerose,
NY 11426. U.S.A.
All nonhomeomorphic simple connected graphs with n edges are constructed through n = 7. In the last section, lower and upper asymptotic bounds on the number of graphs are derived.
1. Introduction Martin Gardner posed the problem of counting the nonhomeomorphic connected structures that can be made with 6 matchsticks in the plane, placing them end to end without intersections [4,pp. 154-1561, and gave the solution, 19 [5, pp. 146, 1501. In graph theoretic language the problem, generalized to n edges and without the geometric and planarity restrictions, is to count the n-edged nonhomeomorphic graphs, the set of which will be denoted H ( n ) . We find I H ( n ) J .precisely through n = 7 using a constructive computer technique, and derive asymptotic bounds for this number using considerations of random graphs. 2. Generation of graphs
The computer method employed was to construct the set C ( n ) of all n-edged graphs (up to isomorphism) as outlined by Heap [7], and reduce this to the set H ( n ) . While it would be preferable to use an orderly algorithm [ l , 81, there is an obstacle to this, namely that such algorithms generate graphs with a fixed number of vertices rather than a fixed number of edges. By generating graphs over a range of vertices and selecting those with the given number of edges this restriction can be overcome, but with a loss of efficiency that may or may not make it worthwhile. It is because either the addition of an edge to or deletion of an edge from a homeomorphically irreducible graph can give a reducible graph that the direct generation of H ( n ) (as opposed to using C(n) as an intermediate) is not feasible by any means. In the construction actually used, the adjacency matrix representation of the graph was used throughout. Two methods can be used to determine if r = r’ from their adjacency matrices A and A ’ : (1) compare A and A‘ to each other, and (2) make canonical matrices from A and A’ and check these for equality. Though the first method is potentially more efficient in a single case, the second is apparently more efficient in this application, i.e. searching to see if a given graph is already included in a list [7, p. 511. 249
250
G. Sorkin
The canonical representation of a graph is found as follows [7, pp. 50-531: Begin with the given adjacency matrix A = A(r). Sort (permute) the rows of A on the basis of their sums, and permute identically the columns of the matrix thus formed. (In effect, label r according to the degrees of its vertices.) Permute those rows and columns having the same sum to maximize A, where we define M > N if the first nonzero element in the upper triangle of the matrix M - N, in standard top-down left-right order, is positive. The permutation algorithm used is one by Johnson and Trotter [3, pp. 2-31 which minimizes the number of rows and columns permuted. The computer program ADJMAT lists C ( n )given C(n - 1). (Heap’s method was non-recursive, but he was working on the basis of the number of vertices in the graphs. Since the notion of a complete graph on n edges does not make sense, and there does not appear to be a way around this using line graphs, there is no comparable method for edges, and recursion must be used.) ADJMAT adds an edge to each graph in C ( n )in every possible way. For each graph formed, the program adds the graph to the list of n-edged graphs previously generated if it does not already appear there. The program REDUCE lists H ( n ) given C ( n ) ,the output of ADJMAT. For each graph in C ( n ) ,it “deletes” all vertices of degree 2 and adds the canonical form of the resulting general graph to a list of nonhomeomorphic graphs if it is not already there. The deletion of a vertex u of degree 2 means that the edges (u, ui) and (u, ui) are removed along with the vertex u, and replaced by a new edge (ui,vi). Of course, ui may become connected to uj by more than one edge, and even to itself (i = j is permitted), but the general graphs formed are only being used to decide the homeomorphism of their originals, which are simple graphs. Since each “reduced graph” formed is homeomorphic to its origional, the number of nonisomorphic reduced graphs is precisely the number of nonhomeomorphic graphs.
3. Results ADJMAT and REDUCE were run through n = 7. The number of graphs produced by ADJMAT was checked against established results [6, p. 2411 and found to be correct. The results of REDUCE are given below:
Edges Nonhomeomorphs
1 213 4 1 1 3 5
I
5 6 I 7 10 20 I 42
4. Asymptotic bonds
To best estimate the number of graphs, we should add estimates for every number of vertices p and the given number of edges n ; unfortunately, if p is too
The enumeration of nonhomeomorphic graphs by edges
25 1
small compared to n, these estimates are hard to compute. But because the number of graphs is far greater when p is large, a good lower bound is the number of graphs for a large p which is virtually certain to produce a connected graph with no vertices of degree 2 when n edges are randomly chosen on the vertices. For then we will have a set of graphs which are almost connected and homeomorphically irreducible, and the set must have at least
graphs. It has been proved elsewhere [2] that for a random graph I'p,n(p) on p vertices with n ( p ) edges, if 4 P ) = $P 1% P + YP + O(PL
rp,n(p) becomes
connected with probability approaching 1 as y increases. Fixing n(p) = 2p log p for the duration of the paper makes rp,n almost certainly connected . If a vertex u in rp,n has degree 2, there is a way of choosing all but 2 of the remaining vertices so that 2, is not connected to any of those chosen. Since the probability that u is nor connected to a given vertex is c = 1- n/(S), and this probability declines if we know that u is not connected to some other vertex (leaving less places for the n edges to be), the probability that a given vertex in I'p,n has degree 2 is less than (p;1)cp-3.Using:
the probability that any vertex in
rPsn has degree
2 is less than:
So a graph rp,n(p) is connected and homeomorphically irreducible with probability 1- o(1). Then
GP)" - log n ! p! -n(2logp)-nlog
- 2n log n - n log n.
-
(3 (9 -
n log n--
-plog -
n log n 2 log n
252
G . Sorkin
Also, every graph in H ( n ) , because it is connected, can be constructed by starting with one edge, connecting a second to this, connecting a third to either or both of the first two, and so on. Before the kth edge is added there are at most k vertices in the graph so far constructed, so there are less than ("I) ways of adding the kth edge. Then:
Thus
References [l] C.J. Colbourn and R.C. Read, Orderly algorithms for generating restricted classes of graphs, J. Graph Theory 3 ( 1 079) I X7-195. [2] P. Erdijs and A. Rknyi. On the evolution of random graphs. Bull. Inst. Int. Statist. 38 (1961) 343-347. 131 S. Even, Algorithmic Combinatorics (Macmillan, New York: Collier Macmillan, London, 1973). [4] M. Gardner, Mathematical games, Sci. Am. (February 1962) 150-161. [ 5 ] M. Gardner, Mathematical games, Sci. Am. (March 1962) 138-153. [6] F. Harary and E.M. Palmer, Graphical Enumeration (Academic Press, New York, 1967). [7] B.R. Heap. The production of graphs by computer, in: R.C. Read, Ed., Graph Theory and Computing (Academic Press, New York, 1072) 47-62. [8] R.C. Read, Every one a winner, Ann. Discrete Math. 2 (1978) 107-120.
Annals of Discrete Mathematics 9 (1980) 253-257 @ North-Holland Publishing Company
OPTIMUM RESTRICTED BASE OF A MATROID MA Chung-fan, LIU Chen-hung and CAI Mao-cheng Znstitute of Mathematics, Academia Sinica, Peking, The People’s Republic of China
1. Problem statement Let M = (E, I) be a matroid with the rank function R (where E = {el, . . . , en} is a set of elements and I is the collection of independent subsets of E). Let l ( e ) ,the weight of e, be a real function on E and let P={X,, Xz, *
*
3
Xm}
be a partition of E. Assign two integers base T of M satisfying ai s IT f l X i
I s bi
a, and bi, with 0 =sa, 4 bi to each &. A
( i = 1,2, . , . , m )
is called a restricted base of M. A restricted base of M is called minimum restricted base if it has the minimum weight among all restricted bases of M. Good algorithms are known for finding an optimum base of a matroid or an optimum intersection of two matroids. Here we propose a good algorithm for finding a minimum restricted base of a matroid. Using linear programming duality and Edmond’s result [2] we can easily obtain the following theorem.
Theorem. Let M = ( E , I ) be a matroid, l ( e ) a weight function on E, T a base of M, P = { X I ,X,, . . . ,X m } a partition of E and let P ( T )= {A, B,C, D, I, H } be the following partition of P A = { X i I I T n X i I = a i < b ia n d X i E P } , B = { X i I I T n X i I = b i > a i and X i ~ P } , C = { X i I I T n X i I < a i and X i € P } , D = { X i I J T n X i ( > bai n d X i E P } , I = { X i I a i < I T n X i I < b i and X i ~ P } , H = { X i I I T n X i ) = a i = b ia n d X i E P } . Then T is the minimum restricted base of M if and only if the following conditions hold: (1) C = D = g , 253
254
M a Chung-fan
( 2 ) There exist non-negative ai and ai > 0
+ Xi
EA
UH U
et
al.
pi such that aipi= 0 for i = 1 ,2 , . . . , m and
C,
pi > 0 3 xi E B u H u D, ( 3 ) T is a minimum base of M with weight function l'(e), defined by
l'(e) = l ( e )- ai + p i
for all e E Xi, i = 1 ,2 , . . . , m.
The proof is omitted. 2.
An algorithm
The main idea of the algorithm is to gradually change ai, pi and T, while preserving the validity of conditions (2) and (3). As T changes, the sets C and D gradually become smaller and smaller until they become empty. Then by the theorem T must be a minimum restricted base.
Step 0 : Let a:= pp= 0 for i = 1 , 2 , . . . , m and let l'(e) = l ( e ) - a : + p y for e E Xi. Using the greedy algorithm a minimum base T o of M with weight function I'(e) and the corresponding partition P(T') = {A', Bo, c",I",H"}of P are found. Obviously a:, pp and T o satisfy condition (2) and condition (3). In general, set I k ( e ) =l ( e ) - a k + p : for e E X i . Let T k be the minimum base of M with the weight function l k ( e )and let P ( T k )= {Ak,B k , Ck,D k ,Ik,H k } be the partition of P with respect to T k .For X i EP we have crk20, pk30 and akpk= 0 and a;, pk and T k satisfy conditions (2) and (3). Step 1: If Ck= Q, go to Step 2; otherwise, choose any X i , E C k . (1.0) Let L = N = Q, X i , is labelled (0). If Xi, E T k , stop, no solution exists; otherwise, each element in ( X i ,- T k ) is given the label (0, 8). (1.1) Let L and N be the sets of all labelled sets X i and all labelled elements e respectively. According to the principle of first labelling-first scanning we scan each Xi of L as follows: For each element e, in ( X i - T k )set Y ( e , )= {e I e E %(e,) and e$: (Ux,ELX & ) } , where %(e,)denotes the cycle in T kU { e , }
a(et)=
[
if U e , ) =@I, max l k ( e ) otherwise, :(etl
A(e,)= lk(e,)-6(e,),
A ( X i ) = min e, E (Xi- T k)
A = min A ( X i ) . X,€L
A(e,),
255
Optimum restricted base of a matroid
If A = cc., then no solution exists. If A
=
'
&:+A
if X i € L, otherwise.
{a!
Let Z = { X i I X i € L and A ( X i ) = A }
E ( X i )= {e, I e, E ( X i- T k )and A ( e , ) = A } for Xi E Z ,
F(e,)= { e I e E Y ( e , ) and l k ( e , ) - l k ( e )= A } for e, E E ( X i ) , F(Xi)=
u
F(e,) for X i € Z ,
e,eE(X,)
F=
U
Xt€Z
F(Xi)=
U
X,EZ
(
U He,)).
e,eE(X,)
According to the principle of first labelling-first scanning we scan each element in F as follows. Assume that ej E F(e,)c F ( X , )c F and e, has label (p, e,), then ej is given the label ( p + 1, e,) (note that if e j E F ( e , ) n F ( e , )and e, and e, have been labelled (p, e,) and (v,eu), respectively, and v < p , then ej is given the label ( v + 1, e,) by the principle of first labelling-first scanning) and X , is given the label (p 1) (clearly X had not been labelled) and all elements in X , - T k are labelled (p 1,ei). Let L1and Nl denote the sets of all newly labelled rC, and all newly labelled elements respectively. Let N = N U N , and L = L U L , . (1.2) Let 9= N f l T k . (1.2.0) If 9 = 8, return to (1.1). Otherwise, choose any e, E 9. Assume that e, E X i . If Xi E I k U D k U B k , go to Step 3; otherwise, put 9= 9- { e , } and return to (1.2.0)
+ +
Step 2: If D k = 8, stop; T k is the minimum restricted base. If D k# 8, choose any Xi, EDk. (2.0) Set L = N = $3. Xi, is labelled (0), and all elements in Xi,n T k are labelled
8). (2.1) Let L and N denote the sets of all labelled X i and all labelled elements e, respectively. According to the principle of first labelling-first scanning we scan each X i of L as follows: Let
Y ( e , )= { e I e E O ( e , ) and e #
U
X,tL
X,}
for all e, E X i
Tk,
Ma Chung-fan et al.
256
where O(e,) denotes the cocycle in (E - T k )U{e,}. if Y ( e , )= 8, I k ( e ) otherwise,
A(e,)= S(e,)- lk(e,), A
= min
A ( X i )= min A(e,), e, E XInT k
A(Xi),
XIEL
and min
if L n A k = @ , a: otherwise.
If 8 = A = 00, stop (no solution exists). If 8O and a:<e, set p ; = f l - a : , a;=O. (Obviously X i € H k . ) If a:>O and a : a 8 , let a f = a f - O . Using updated a: and p: modify I k ( e ) and then go to (2.2). If 8 a A , for each X i in L, we modify a: and p: as follows: If p:aO and a:=0, set p:=p:+A. If a;> 0 and a: C A, set p: = A -a:, then af= 0. (Obviously Xi E Hk.) If a:>Oand a:z=A, let a : = a : - A . Let Z = { X i l X i ~ L and A ( X , ) = A } ,
E(Xi)= {e, I e, E X i r l T k and A(e,) = A}, X i E Z , F ( e , ) = { e l e E Y ( e , ) and I k ( e ) - l k ( e , ) = A } ,
U
F(Xi)=
F(et),
e, EE(XI)
F=
U
X,EZ
F(Xi)=
U
(
U He,)).
X ~ E Z e,EE(X;)
According to the principle of first labelling-first scanning, all elements in F are labelled as follows: Assume that e, E F(e,) E F ( X , ) c F and e, has had label (p, eq),then ej is labelled (p + 1,e,). X, is labelled (p + 1)and all elements in X , n Tkare labelled (p 1,ej). Let L1 and N, denote the sets of all newly labelled X i and all newly labelled elements e respectively. Put L = L U L1 and N = NUN,. (2.2) Let 9 = N - T k . (2.2.0) If 9= @,return to (2.1); otherwise, choose any e, E 9, say e, E Xi. If X i E Ikor X i E A k and a: = 0, go to Step 3; otherwise, set 9 = 9 -{es} and return to (2.2.0).
+
257
Optimum respicfed base of a mafroid
Srep 3: (3.1) Assume that e, has label (r, eJ, using backtracking method we can find the sequence of labels:
(0,8), (1, GI, (1, ejl), . . . ,( r - 1, etJ, ( r - 1, ej,J (r, rr,h Let w={er,, ei,, ern,ei2,.. . , e,_,, ej,-l,e,, es}. Set T k + * = ( TU k r ) - ( T k nr). (3.2) Determine the partition P(Tk+')of P with respect to Tk+'. (3.3) Let a:+' = a; and p:+' = p; for i = 1 , 2 , . . . , m. Set k = k + 1 and return to Step 1. The proof of the algorithm will be published in detail in Scientia Sinica.
Remark 1. From the definitions of A and 8, after modifying a:, p ; and Tk,we have Ck+' c_ Ck, Dk+' c_ Dk, and conditions (2) and (3) still hold. Remark 2. After T kis changed into Tk+'it can be shown that Tk+'is a base of M,and a:+', pf+' and Tk+'also satisfy conditions (2) and (3). Remark 3. It can also be shown that if A = 00 at Step 1, or A
= 8 = CQ at Step
2,
there is no solution.
References [1) [2] [3] [4]
E.L. Lawler, Combinatorial Optimization: Networks and Matroids (1976). Edmonds, J. Math. Programming 1 (1971) 126-136. M. Iri and N. Tomizawa, J. Operations Res. SOC.Japan 19 (1976) 32-57.. Ma Chung-fan, Liu Chen-hung and Cai Mao-cheng, Optimum restricted base of a matroid, Sci. Sinica, to appear. [5] Y.J.Chu and C.H.Liu, On the shortest arborescence of a directed graph, Sci. Sinica 14 (1965) 12961400.
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Annals of Discrete Mathematics 9 (1980) 259 @ North-Holland Publishing Company.
A CHARACTERIZATION OF NON-HAMILTONIAN GRAPHS
WITH LARGE DEGREES Roland HAGGKVIST University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Abstract Using standard notation we put S(G) = minimum vertex-degree in G, v(G) = cardinality of the vertex-set in G, ar(G)=cardinality of a maximum stable set in G (i.e. the maximum number of pairwise non-adjacent vertices in G).
Sample Theorems. (1) Let G be a non-hamiltonian 2-connected graph with S>$(v+2). Then G contains a stable set of four vertices with at most a ( G ) - l neighbours altogether. (2) Let G be a non-hamiltonian 2-connected graph with S a i ( v + 2 ) . Then G contains a stable set of [i(v+lo)] vertices with at most $(v- 1) neighbours altogether. (3) Let G be a non-hamiltonian 2-connected graph with tSa&v. Then G contains a set of rn 3 z v vertices whose deletion leaves a graph which cannot be (vertex-) covered by rn paths.
259
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Annals of Discrete Mathematics 9 (1980) 261-264 @ North-Holland Publishing Company.
UN PROBLEME D'EXTREMUM DANS LES ESPACES VECTORIELS BINAIRES J. WOLFMANN Uniuersiti de Toulon, UER de Sciences et Techniques, 8 3 / 3 0 La Garde, France Let IF, be the Galois field of order 2 and k a positive integer. We consider the following problem: find the minimum cardinality of a set E, E c F'& such that E contains, at least, half the points of each affine hyperplane. This problem is related to coding theory. We give some results and conjecture and, also, a new characterization of difference sets in 2-elementary abelian groups.
1. Introduction Soit [F, le corps fini B 2 ClCments et k un entier positif. On considbre, dans cet article, le problbme de la dktermination de sous-ensembles de F[,: vCrifiant une certaine propiCtC et dont le cardinal est minimum. I1 traduit en termes purement gComCtriques des questions relatives au code de Reed et Muller d'ordre un et permet une caractkrisation nouvelle des ensembles B diffbrences de [F; (pour k pair). Le vocabulaire et les dCfinitions concernant les codes ne sont pas les plus mieux i faire comprendre le lien entre les habituels; ils sont choisis de faGon ? diffkrentes notions concernkes sous leurs aspects gkomCtriques.
2. Le probleme Propriete 2.1 (P). Soit IFz le corps a deux iliments et k un entier non nul. O n considtre la propriiti suiuante: on dit que E clT-5 posstde la propriiti (P)si
()'
Pour chaque hyperplan afine, E contient au moins la moitii des points de cet hyperplan.
I1 existe de tels ensembles qui peuvent Ctre construits B partir d'hyperquadriques (voir [7]). On s'intkresse alors au probkme &ant.
Problbne 2.2. Trouver les ensembles vCrifiant la propriCt6 (P)et dont le cardinal est minimum. 261
262
J . Wolfmann
3. Le theoreme fondamental
3.1. Code de Reed et Muller d’ordre un. (a) Codes line‘aires: Un code linCaire (n, k) est un sous-espace vectoriel de dimension k d’un espace vectoriel fini V de dimension n. On appelle n la longueur du code, k sa dimension et V l’espace ambiant. Ce vocabulaire est utilisC lorsqu’on fait intervenir une certaine distance dans V, la distance de Hamming. En particulier, l’ensemble des parties de 55 est un F,-espace vectoriel V pour 1’opCration diffkrence symktrique qu’on notera + , A + B = (A \ B)U (B \ A ) (la loi externe est triviale). La distance de Hamming sera alors dCfinie par: d(A,B)= (A+ B ( (cardinal de la diffkrence symetrique). (b) Difinition. Le code de Reed et Muller d’ordre un, de longueur 2k, est le code, pour l’espace ambiant prCcCdent, dont les ClCments sont: l’ensemble vide, 5 : et les hyperplans affines. I1 sera dCsignC dans la suite par 9.
Thhreme 3.2. Soit: 9 le code de Reed et Muller d’ordre un de longueur 2k, 8 l’ensemble des parties de IF: dont la distance c? 9 est maximum, JU l’ensemble des solutions du Probkme 2.2, alors: 8 = A+ 9.
4. Consequences 4.1. Rayon de recouvrement Definition. Le rayon de recouvrement d’un code C est le plus petit entier r tel que les boules (pour la distance de Hamming) de rayon r centrCes sur les ClCments de C recouvrent I’espace ambiant.
Le problkme de la dktermination du rayon de recouvrement est en gCnCral non rksolu. On dCduit du theorbme le corollaire suivant:
Corollaire 1 (3est dCfini comme en Thkorkme 3.2). Si r est le rayon de recouvrement de 9 et m le cardinal de chaque ensemble solution du Probltme 2.2 alors: m+r=2k. 4.2. Ensembles it dilferences. Un ensemble i diffkrences d’un groupe abklien fini G est une partie propre D de G telle que le nombre de solutions ( x , y ) E D 2 de x - y = g avec gfO est une constante A indkpendante de g. D est dit trivial si D = { x } ou D=G\{x}.
263
U n problkme d’extrimum dans les espaces uectoriels binaires
Dans le cas ou G = (IT,”, +) (groupe 2-abklien klkmentaire) Mann [5] a trouvk les parambtres des ensembles de diffkrences non triviaux; il existe E E { - 1, 1) tel que:
k = 2t, ID(= Z2’-* + &2‘-l, A = 22t-2+ E2I-l. On distingue des ensembles ii diffkrences positifs ou nkgatifs suivant E , les positifs ktant les complkmentaires, dans 55,des nkgatifs. E n ce qui concerne ce sujet et leur lien avec la thborie des codes on peut consulter [l, 2,8,9]. En utilisant la transformke de Fourier finie (voir [l])on peut montrer que les fonctions caractkristiques des ensembles prkckdents sont les fonctions courbes (voir [6]) et que ces ensembles sont ceux dont la distance I? 3 est maximum (voir [4]). Le thkorbme a donc le corollaire suivant qui donne une nouvelle caractkrisation des ensembles 21 diffkrences de (IF;, +).
Corollaire 2. Les ensembles h difirences postis du groupe (55, +) (k pair) sont les ensembles solutions du Problkme 2.2.
5. Cas oi la dimension est impaire
Lorsque k = 2t+ 1 le rayon de recouvrement de 3 n’est pas connu et les solutions du Problbme 2.2 encore moins. Malgrk tout il existe des bornes pour ce rayon (voir [3]) ce qui conduit ii la:
Proposition. Soit k = 2 t + l et m le cardinal de chaque ensemble solution d u Problkme 2.2 alors: 22t+ (Jz)2t-’ < m G 221+ 2‘. La considkration du cas k = 2t et d’ensemble vkrifiant (P) pour k conduit B la conjecture suivante
= 2t
+1
Conjecture. Si k = 2 t + 1: m = 22’+ 2‘.
6. Conclusion La dktermination du rayon de recouvrement du code de Reed et Muller d’ordre 1 pour k = 2 t + 1 reste un problkme ouvert. I1 en est de m$me de la classification et du dknombrement des ensembles ZI diffkrences des groupes 2-abkliens klkmentaires. La mise en kvidence, dans cet article, du lien entre ces questions et le Problbme 2.2 devrait fournir une voie nouvelle de recherche. Le Problbme 2.2, pour k = 2t + 1:voir la session de problkmes dans ce volume.
264
J. Wolfmann
References [l] P. Camion, Difftrence sets in elementary abelian groups, Stminaire de Mathtmatiques Sup&ieures, 1978 (Presses de I’Universitt de Montrkal, Montrtal, Qut., B paraitre). [2] J.F. Dillon, Elementary Hadamard difference sets, Thesis, Univ. of Maryland (1974). [3] T. Helleseth, T. Klove et J. Mykkeltveit, On the covering radius of binary codes, IEEE Trans. Information Theory 24 ( 5 ) (1978) 627-628. [4] F.J. MacWilliams et N.J.A. Sloane, The Theory of Error Correcting, Codes (North-Holland, Amsterdam, 1977). [5] H.B. Mann, Difference sets in elementary abelian groups, Illinois J. Math. (1965) 212-219. [6] O.S. Rothaus, On “Bent” functions, J. Combin. Theory 20 (A) (1976) 300-305. [7] J. Wolfmann, Codes projectifs B deux ou trois poids associts aux hyperquadriques d’une gtomttrie finie, Discrete Math. 13 (1975) 185-211. [8] J. Wolfmann, Codes projectifs B deux poids, “caps” complets et ensembles de difftrences, J. Combin. Theory 23 (A) (1977) 208-222. [Y] J. Wolfmann, Aspects gtomttriques et combinatoires de Etude des codes correcteurs, Thbse, Universitt de Paris 7, Paris (1978).
Annals of Discrete Mathematics 9 (1980) 265 @ North-Holland Publishing Company.
ON MINIMAL NON-HAMILTOMAN LOCALLY HAMILTONIAN GRAPHS
C.M.PAREEK and Z. SKUPIEN Uniwrsity of Waterloo, Waterloo, Ontario, NZL 3G1, Canada
Abstract A graph G = (X, E) is called Harniltonian if it has a Hamiltonian circuit, i.e. a circuit whose set of vertices is the vertex set X of G. G is called locally Hamiltonian if for every x in X the graph exists and is Hamiltonian. In this note, we show that a minimal non-Hamiltonian, connected locally Hamiltonian graph is of order eleven.
265
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Annals of Discrete Mathematics 9 (1980) 267-268 @ North-Holland Publishing Company
ON A FAMILY OF SELFCOMPLEMENTARY GRAPHS* Sergio RUIZ Instituto de Matemciticas. Universidad Catdica de Valparaiso, Valparaiso, Chile “Dedicated fo Professor Roberfo Frucht on his 73rd birthday, August 1979” Let X be a selfcomplementary graph, 9 its complementing permutation and F = E V ( X ) } a family of graphs such that y, = Yrcx,for all x E V ( X ) .Then the generalized X-join of F is a selfcomplementary graph.
{ Y, 1 x
The following construction of selfcomplementary graphs is an attempt to improve on the author’s previous paper [3]. All graphs considered here are finite, undirected, without loops or multiple edges and V ( G ) ,E ( G ) denote the vertex-set and the edge-set of the graph G respectively. The complement G of G is a graph on V ( G )whose edge-set consists of the edges not in E ( G ) .Further G is a selfcomplementary graph (briefly S.C.graph) if there is an isomorphism 9 (called a complementing permutation [l])of G onto G. In [4], Sabidussi introduced an operation between graphs as follows: The generalized G-join of a family F = { Y, x E V ( G ) }of graphs is the graph J with vertex-set
I
V ( J )= {(x, y) I x E V ( G ) and Y E W‘,N and the edge-set E ( J )= {{(x, Y ) , y ’ ) } 1 {x, x’) E E ( G ) or else x = x ’ and {y, y’} E E(Y,)}. It is easily seen that is the generalized G-join of the family {y, 1 x E V ( G ) } . (XI,
Theorem. Let G be an S.C. graph with complementing permutation 9 and let F = { Y , I x E V ( G ) }be a family of graphs such that YT(,,= q, for all x E V ( G ) . Then the generalized G-join J of F is an S.C. graph. The proof is straightforward: clearly, the permutation 9*acting on V ( J ) and defined by S*((x, y ) ) = (Y(x), y ) is a complementing permutation for J. It follows that every graph G is isomorphic to both the K,-join of { G } and the G-join of {Y, I x E V ( G ) }with Y, = K1 for all x (where K , is the singleton graph). We will say that an S.C.graph is GJ-representable if it is a generalized join distinct from these two trivial cases. *Work done with the financial aid of Direccih General de Investigacih de la Universidad Cat6lica de Valparaiso under the project 01 1-04-78. 267
268
S. Ruiz
The following facts can be easily established: (i) The number II of vertices of an S.C. graph satisfies n = 0 or n = 1 (mod 4) (see, e.g. [5]). Exactly 4 of the 1Os.c. graphs on 8 vertices and exactly 6 of the 36 S.C. graphs on 9 vertices are GJ-representable. (ii) There is an S.C.graph on 12 vertices which has two essentially different GJ-representations. (iii) If all Y, are equal to a given S.C.graph H and if G is also s.c., then the generalized G-join of the graphs { Y,I x E V(G)} is the lexicographic product G[H] which is S.C.by the theorem. As an interesting example, consider the S.C. graph C,[C,]: it is vertex transitive but not edge transitive; its automorphism group is the wreath product of the dihedrical group D5with itself whose order is one million.
Achowledgements Many thanks to Ivo Rosenberg and the referee for their contribution on the stylistic presentation of this work.
References [l] R.A. Gibbs, Selfcomplementary graphs, J. Combin. Theory 16 (B) (1974) 106-123. [2] G. Ringel, Selbskomplementiire Graphen, Archiv Math. 14 (1963) 354-358. 131 S. Ruiz, Construccidn de grafos autocomplementarios por dilatacidn, Sigma: Revista de Matematicas Aplicadas de la Univ. de Chile 3 (2) (junio 1977) 1-7. [4] G. Sabidussi, Graph derivatives, Math. Z. 76 (1961) 385-401. [S] H. Sachs, h e r selbskomplementiire Graphen, Publ. Math. Debrecen 9 (1962) 270-288.
Annals of Discrete Mathematics 9 (1980) 269-276 @ North-Holland Publishing Company
FORMES BILINEAIRES SYMETRIQUES SUR UN ESPACE VECTORIEL DE DIMENSION FINIE SUR LE CORPS A DEUX ELEMENTS: APPLICATIONS AUX MATROIDES BINAIRES Alain DUCHAMP Dtparrement de Mathe‘mafiques, Unioersite‘ d u Maine, 7201 7 Le Mans, France Let M ( E ) be a binary matroid on E and let 8, %*, Ct, = 8 n8* be respectively the cycle space, cocycle space and bicycle space of M ( E ) . The object of this paper is to characterize the rank of go by the existence of particular bases of %. First, this result is used to display the decomposition of E in cycle and cocycle as well as the principal tripartition of E (see [3]). Then we characterize the binary matroids satisfying ‘8 = 8, as restriction minors of self-dual binary matroids, and more generally we show that all binary matroids are minors of self-dual binary matroids. Next, we compute the maximum rank and the maximum cardinality of certain subsets of $-collection of cycles defined by properties of intersection.
1. D66nitions et notations (1) Soient 8 un espace vectoriel de dimension finie sur le corps 1 deux ClCments GF(2), et une forme bilinkaire symCtrique F sur 8. Le noyau de F est
8, = {x E 8 I Vy E 8, F(x, y) = 0) et F est dite non-de‘ge‘ne‘rie (ou 8 non isotrope) si
8, = (0). Si Vx E 8, F(x, x) = 0, F est dite alternde. (2) Etant donni un ensemble fini E, on consid6re S ( E ) ensemble des parties de E, comme espace vectoriel sur GF(2) pour la loi habituelle “diffkrence symitrique”, notie +, et ( , ) la forme bilinCaire symitrique sur 9 ( E ) difinie par: VX, Y t E, (X, Y) = 0 ou 1 suivant que le cardinal de X n Y est pair ou impair. On note X =(x,x). (3) Pour un matroTde binaire M ( E ) sur E, 8 (resp. 8*, 8, = 8 n 8 * ) dCsigne le sous-espace (dans B(E)) des cycles (resp. cocycles, bicycles) yP* est l’espace orthogonal 1 8 pour la forme ( , ) et 8, son noyau. Pour G c E , M\G= M x (E- G) (resp. M / G = M,(E - G)) dCsigne le matroi’de sur E - G obtenu en supprimant (resp. contractant) les ClCments de G. On dit que M (ou 8)est biparti (resp. eulirien, cyclique, cocyclique, autodual) si X = O , V X E ~(resp. X = O VXE%*, 8*c%, $c$*, %=8*).On note k = d i m 8 , n=dim$* et q=dim8,. 269
270
A. Duchamp
(4) Considtrant un matroi'de binaire M ( E ) et une optration interne commutative, P, non ntcessairement partout dtfinie sur GF(2) on dit que 9 ~ vtrifie 8 P ou est une P(n)-famille de cycles de M, si vX,Y~9,X#YJxPPddefiniet~PP=(X,Y).
Les lois considtrkes sont nottes symboliquement n. Par exemple une P2-famille avec p2= (0 no = 0 , o n 1 = 1, 1 n 1 = 0 ) signifie que pour X, Y E 9 X # Y . S ( X , Y)=X+T.
2. Caracterisation des formes non ddginerCes (voir aussi [l]) ThCorhme 1. Soit F une forme biline'aire syme'trique sur un espace vectoriel de dimension finie k a 1 sur GF(2). F est non de'ge'ne're'e si et seulement si il existe une base ( x i : i E I ) de 8 ue'rifiant (i) pour F non alterne'e F(xi, xi) =
si i = j , 0 si i f j , 1
i, j E I,
(ii) si F est alterne'e
k est pair et F(xi,x i ) =
0 si i = j , 1 si i f j ,
i, j E I.
La dkmonstration s'effectue par rtcurrence sur k 2 l'aide du lemme:
Lemme. Si F non de'ge'ne're'e et k 3 3 il existe x l , x2 E 8 tels que i=1,2+F(xi,xi)=0
et F ( x , , x , ) = l .
Etant donne une base B = (q:i E I ) de 8 soit A = [F(ei,ei)]i.i.I la matrice de F par rapport 2 B. A est dite alternte ou non selon E On a alors rang A +dim go= dim 8. Si B' est une base de 8 et P la matrice de passage de B 2 B' (matrice exprimant les eltments de B' par rapport 2 ceux de B) la matrice de F par rapport 2 B' est A'='PB P oh 'P dtsigne la transposte de la matrice P. I1 rtsulte de ceci et du Thtorbme 1:
Theoreme 2. Soient F forme biline'aire syme'trique non alterne'e (resp. alternke) sur 8, k =dim 8, q =dim go et A = [aij] une matrice carre'e en (0,1), d'ordre k, syme'trique de rang k - q , non alterne'e (resp. alterne'e). Alors il existe une base
Applications aux matroi'des binaires
271
(x,: i E I ) de 8 vtrifiant
F(xi,xi) = aii Vi, j
E I.
(En outre si F alterne'e k - q est pair). 3. Bases de cycles rhmarquables et tripartition d'un matroide binaire
(1) En dimension infinie le rCsultat du ThCorkme 1 est en d6faut.
Exemple. Soit 'i4 l'espace des cycles engendrk dans N par (Cl: i EN) oh Co= {0,1}et pour i k 1, Ci = {0,2, i + 2). 8 est l'espace des cycles du graphe:
(2) On considkre les propriCtCs P ( n ) suivantes: Po= ( o n o = 0 , on 1 = 0 , 1n 1= o), p1= ( o n o = 1, on 1= 0, 1n 1= o), p2 = ( o n o = 0, on 1= 1, 1n 1 = o), p3 = (ono= 0, on 1 = 0, 1n 1= I), Qo=(OnO=O),
Q $ = ( l n l =l),
Q1=(OnO=l),
QT=(lnl=O).
( o n o = 1, on 1= 1, 1n 1= I), P:= ( o n o = 1, on 1= i , 1 n 1= o), P: = ( o n o = 1, on 1= 0, 1n 1= I), P; = (ono = 0, on 1= 1 , i n 1= I),
P$=
Etant donni une P-base de cycles '& = (Ci: i E I) d'un matro'ide M ( E ) le tableau 1 donne la valeur de q = dim gosuivant P. Pour simplifier on suppose k = dim 8 k 1. On note I. = { i E I 1 Ci = 0}, I, = I- I,. 1 x1dCsigne la cardinal de l'ensemble X. (3) Du tableau 1 et du ThCorBme 2 dCcoule alors: Soient M ( E ) un matro'ide binaire, % = ( C i :i E I) une base de cycles, k = dim 8, q =dim 8,.
Proposition 3. O n suppose k k 1. Les proprie'te's suivantes sonr e'quivalentes d M non biparti et q = 0. (1) 3%ve'rifiant QT. (2) 3% ve'rifiant f l avec 111( = 1. (3) 3% ve'rifiant PT avec F1= I. (4) M non biparti et 3% vkrifiant P1avec To= 0. si en outre E = o
272
A. Duchamp
( 5 ) 3% virifiant P r avec JIoI= 1. Si en outre E = 1 (6) 3% virifiant avec [Ill= 1. Tableau 1
Pol # 0
Restriction
io=o 4
0
Pol
io=1
il= 1
1
1101- 1
: p
Qo
: p
IZ,)# 1
Ilol # 0
#0 i o = iet
I;=o
1 11,1-1
sinon 0
QX
Q:
k-1
0
E=O 0
k-2
4
QI
#0
io=o IIJ- 1 i(,= 1 1111
Relation
1111
q=E
Relation
Restriction
#0
k-1
Pol
P
1111
i,= o
q=E+l
k
E= 1 1
q=E
Proposition 4. On a iquivalence de ( 1 ) M biparti e f q = 0, (2) E = 0 et 3% virifiant Q,. Proposition 5. O n suppose k 2 1. Les propriitis suivantes son? iquivalentes a M non biparti et q = dim go. ( 1 ) 3% virifiant Po et I , # @et (IoJ=q. (2) 3% virifiant P$ et [I1[ = q + 1. Si en outre E = ~f (3) 3% virifie P2 et I , # 9 et lIol= q. Si en outre E = i j + l (4) 3% uirifie P: et ~ I ~ q +JI = . Proposition 6. On suppose k 3 1. O n a iquivalence de ( 1 ) M bipartie et q = dim go, (2) 3%= (C, : i E J + K ) base de cycles auec J f l K = 9, telle que: (i) %=(C,: ~ E J virifie ) Qo er JJI=q, (ii) 129 = (C,: i E K) virifie Q1 et I? = 0, (iii) 2l est orthogonale h 129 (pour ( , )). (4) Tripartition principale d'un matroide binaire [4].
Applications aux matroLdes binaires
Etant donnC un matroi'de binaire M ( E ) toute partie A c E orthogonale dkcompose en:
273
gose
A = y(A)+ @(A) oh y(A) E 8 et @(A)E 8". (a) Supposons M non biparti et soit V = (C,: i E I) une Po-base de cycles de M, Io={iEIl et I 1 = I - I o . Pour A C E notons
ci=O}
I ( A )= { i ~ I I (Ci, A ) = 1).
Proposition 7. Pour A c E orthogonal d go (I(A)c I]) Y(A= ) C Ci [i E I(A)I
est un composante cyclique de A.
Corollaire 1. (P, Q, R) e'tant la tripartition principale de M R = UC, [iEIo], P={~EE-R
Q = {X E E - R
i)=y(~)-~,
I I(x)=o)= ( E + y ( ~ ) ) R. -
Et un des rksultats de [2]:
Corollaire 2. Si k = dim 8,q = dim go,
y(E)+q=E. (b) Supposons M biparti et soit c& = (C,: i E J + K ) une base de cycles satisfaisant la Proposition 6 et pow A c E notons
K ( A )= { i E K I ( Ci, A) = 1).
Proposition 8. Pour A c E orthogonal d go ( K ( A )c R ) notons si K ( A )= 0,
y(A) =
si K ( A )= 1, y(A) =
c C, c
[i E K ( A ) ] , Ci [ i E K - K ( A ) ] .
Alors y(A) est un composante cyclique de A. 4. Matroide cocycliques et matroides autoduaux Des propositions 3 et 4 dCcoulent:
Proposition 9. Soit M ( E ) cocyclique et B une base de M. On note r ( M ) = n , r(M*)= k.
274
A. Duchamp
Si M est non eulkrien (resp. eulirien) il existe n - k cocycles linkairement inde'pendants vkrifiant QT = ( 1 n 1 = 0) (resp. Q1= (0 n0 = 1)) et i n c h dans B.
Proposition 10. Soit M ( E ) matroi'de binaire. Alors M cocyclique kquivaut a: I1 existe un matroi'de binaire N ( E + F ) autodual tel que (i) si M eule'rien: r(N)= r(M)+1, M = N ( E ) et F circuit de N, (ii) si M non eule'rien: r ( N )= r ( M ) et M = N ( E ) . Proposition 11. Soit M ( E ) un matroi'de binaire, r ( M ) = n, r(M*)= k. Les proprie'te's suivantes sont iquivalentes: (i) M non biparti et dim 8, = q. (ii) II existe un matroi'de binaire N ( E + F ) cocyclique avec IF1 = k - q > O , F indipendant dans N* et M = NIF. En outre r(N*)= r(M*) et M eule'rien si et seulement si N eule'rien. Proposition 12. Soit M ( E ) un matroi'de binaire, r ( M )= n, r(M*)= k. Les proprie'te's suivantes sont e'quivalentes: (i) M biparti et dim 8, = q. (ii) II existe un matroide binaire N ( E + F ) cocyclique avec IF)= k - q + 1 impair, F cocircuit de N et M = NIF. En outre r(M*)= r(N*). Corollaires. (i) Tout matroi'de binaire est le contracte' d' un matroi'de cocyclique. (ii) Tout matroi'de binaire est sous-matroi'de d'un matroi'de cyclique. (iii) Tout matroi'de binaire est mineur d'un matroi'de autodual.
5. Determination de max-rang (9) et max-card (W pour 9 P-famille de cycles Soient M ( E ) un matroi'de binaire et 9 c 8 une P-famille de cycles. On note dim 8 = k, dim go= q. 9 est dite Paire (resp. Impaire) si
VXE 3
X =o
(resp. X = 1).
Pour Bliminer les trivialitks on suppose toujours que k 2 1 et sauf spCcification contraire que M est non biparti. [ x ] dCsigne la partie entikre (par dBfaut). ri = 0 ou 1 suivant que n est pair ou impair. Le tableau suivant donne alors la valeur du rang maximum et de la cardinalit6 maximum d'une famille 9 de cycles, Paire, Impaire ou vCrifiant P. Renvois concernant le tableau 2 (1) M biparti, (2) M non biparti,
N
a B
Y
Applications aux rnatrordes binaires
Y
N
u
I
w
2
x
k
b
N
Y
N
YI
-
N
Y
N
Y
I
-
u
Y, N
t”
s
-
N
v
Y
t”
I
*
--9
Y
I
U Y
h 4
U
+ +
Y v -1-
w
h
+
5
-IN
Y
Y
I
3
Y
Y
M
5 2
275
A . Ducharnp
276
(3) (i) si M biparti et q = 0 max-rang(9) = k, max-card(9) = k + 1,’ (ii) si M non biparti et q = 0 max-rang(9) = k - 1 , max-card(9) = k - I;, (4) 1 +2C(k+q-1)/21 si k + q pair, max-card 9 =
(5)
1.
2C(k+q-1)’21 si ( q f 0 et k + q impair) ou (q = O et k impair s7),
max-card 9= k - q k-q+l
si q = 0 et k impair d 5, si q = O ou k - q impair, si q # O et k - q pair.
Remedements Que Messieurs I.G.Rosenberg et le referee veuillent bien trouver ici l’expression de ma gratitude, le premier pour avoir suscitC cette Btude et le second pour ses utiles conseils et r6f6rences concernant les ThCorBmes 1 et 2 et la transcription, dans le langage de la thhorie des codes correcteurs binaires [3], des rksultats exposb.
Bibliographie [I] J. DieudonnB, La gkombtrie des groupes classiques (Springer, Berlin, 1963). [2] H.de Fraysseix, ProprittCs de paritt des bases d‘un matroide binaire, C.R. Acad. Sci. Paris 286 (A) (1978)1171-1173. [3] F.J. MacWilliams and N.A. Sloane, The Theory of Error-correcting Codes (North-Holland, Amsterdam, 1978). [4] R.C. Read and P. Rosenstiehl, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978)195-226.
k
En appliquant ce rCsultat au matroide M ( E ) de rang 1 sur un ensemble fini E de cardinal n (d‘oii - 1) on retrouve une proposition de I.G. Rosenberg (communication privbe): Soient des ensembles finis A,, . . . , A , , tels que Ai = O V i et Ain A , = 1, V i , j , i # j. Alors
= dim % ‘ =n
lUA,Is2q.
Annals of Discrete Mathematics 9 (1980) 277-282 @ North-Holland Publishing Company
ON THE MAXIMUM VALUE OF A QUADRATIC FORM OVER BINARY SEQUENCES Victor G. TUPITSYN Department of Mathematics. Mary Washington College. Fredericksburg. VA 22401, U.S.A.
Let {En}be the set of all binary sequences of length n, i.e.:
. . . , E,,)},
E ~ ,
where
E~ = 0
or 1,
1s i s n .
Consider the quadratic form:
In this note we determine the sequence in {En}for which the quadratic form (1) assumes its maximum value. We shall apply this result to the study of subgraphs of the complete bipartite graphs. It is easy to verify that:
Lemma 1. The maximum value of (1) on {En}occurs at the two sequences (of length n ) : (010101 . . . 0101 . * .)
and
(101010 *
* *
1010
. .).
Proof. We shall prove Lemma 1 by induction. First, denote by {En.k+l} the subset of {En}consisting of all binary sequences of the form
, . . . , F ~ - , ,1-8.8, 1-8,8, 1-8 , . . .), where 8 = 0 or 1. Hence {En)= {En.,,+,}and (1 - 4 8 , 1 - & 8 , 1 - 8,. . .) E {En,l}.Second, denote by f k + I ( E n ) = f k + l ( E l , E ~ .,. . , F , ) the expression
2
j=k+l
Then:
c 2i(Ei
j-1
2-’
i=l
- Cj)2.
278
V.G. Tupitsyn
(1). Let us suppose that f k + 1(En)
f k + 1 (En,k
+I),
1.e.: fk+l(&I,
...
&23
7
&n) C f k + l ( E l ,
82,
...
7
Ek-1,
1- 8, 8,1- 8,8,1- 8,. . .). (2)
(2) (The first induction step.) Let us set k = n - 1. It is easy to verify that
(3) We shall prove that Eq. (2) yields fk
(En)sfk
(3)
(En.k).
First, let us remark that:
where summation over { E n . k + l } is denoted by ci$+l.Now, consider the difference: fk(ERk)-fk(En). Using (a), (b) and (c), we get: - fk (En)
f k (En,k)
+ zk-'
= f k + l(En,k+
2' 2-'[( 1
*
1)
-fk+ l ( E n ) +
- 8 - &j)2-
(&&I
- (Ek- L - 81'1
- &j)2].
j=k+l
Since f k + l ( E R k + l ) - f k + l ( E n ) ~ O cases: (A)
&k-1=
(B)
&k-l=6.
(because of (l)),then it remains to consider two
1-8 ;
Now (A) gives us: 1-
(Ek-1-
f'2'[(1
8 ) z -0; - 8-
Ej)2-(&k-1
-&j)2],0a
j=k+l
Using the above, we obtain Eq. (3). Setting & & I = 8, we have:
g 1 -(&k-l-8)2]=+
On the maximum value of a quadratic form over binary sequences
and
- 2k-I j-k+l
1 (-1y
n-k+l
=
m=l
1 1 (- l y k + l 1 3 -. 2 - m = --+-.3 2n-k+l 2' 3
Now, using the above, we obtain Eq. (3). Let us set k = 1. This yields fl(En)
s fl (En, 1 )
or equivalently,
1
f , ( ~=)
2-li-;l . ( E i - E ~ ) ~ = S ~ ~ ( E=, ,f~ l) ( i- e, 8, 1- e , 8 , . . .I.
ISirjGn
The proof of Lemma 1 is now completed. Using Lemma 1, we get: n
n
In particular, it is easy to show that:
The following lemma is proved similarly and we omit the details.
Lemma 2. Consider the system:
f i=l ;=I
2-'i-;'Ei(1-&;);
fq=S i=l
Suppose that S s [in]. Then the maximum of
(S>l).
279
V.G.Tupitsyn
280
and
where d is the smallest nonzero integer satisfying the equation: n = d ( S - 1)+1+1 + m
with O S l s d - 1 ;
Il-mISl.
OSmSd-1;
The case S = 1 is trivial.
In particular, setting S = [ i n ] , we obtain: d = 2. This'yields: l = 0 o r 1;
m = l - 1 = 1 o r 0 (when n iseven)
1 = 0 or 1;
m = l = 0 or 1 (when n is odd).
and
So, Lemma 2 is a generalization of Lemma 1. Note that in the case S > a n ] we can get two complementary sequences EE,jS and EE.2'. Now let B = llbijlln be the adjacency matrix of the bipartite graph f,,,where f,, is a subgraph of Kz,,, (2 + w = n ) . Let T(fJ = T ( B )denote the following function:
Consider the transformation:
where
is a permutation on 1 , 2 , . . . , n. As is known, a permutation u can be represented by a permutation matrix IlC,ll,,, where C, = 0 or 1. Using the above, we get:
=
c {2-li-jl ( i 1.1
where
E~
=
C;=,+,
m=z+l
Cmi)( I=l
CIj)}=
c 2+j'Ei(l
- Ej)'
1.1
Cmi= 0 or 1; 1-El = Cf=, Clj= 0 or 1;
CY=l E~ = w.
(7)
281
On the maximum ualue of a quadratic form ouer binary sequences
We now apply the result of Lemma 2 to study of the function (6). This yields the following theorem:
Theorem. Let {(T} be a permutation group (on 1 , 2 , .. . , n ) and let Then
?, c Ks.,-s.
TJf,,) Gfl(ES,,,)= fl(EZs); u = 1,2.
r,,= Ks.,+ I
In the case
we have:
max T, = fl(EE,,) (fl and EE,, are defined above). {d
Corollary. Let
f,,be bichromatic. Then there exists S > O
T(f,,) Sf1(ES,.,), where
u = 1,2.
Example. Let B =Ilbi,llm be the adjacency matrix ZN-k
C
2N-2k+1 bi.i+k
such that
=
i=l
and let
r2N
for k = 1 , 2, . . . , N , for k = N + 1 , . . . ,2N-1.
We have 2N 2N
lrZNl2=C
N
bij = 2
C
(2N-2k
~
+ 1) = 2N2.
k=1
i = l j=1
(Note that = 2N2.) Let us show that there is no u ( u ~ { usatisfying }) u .r 2 N = KNSw Since 6' E {a}, then (repeating (7)) we obtain:
T(u-lKN,,) = Ta-'(KN,N)4 max U%N)
C 2-li-j'q(1 -
2N
E ~ ) ,
where
i.j
.si= N. i=l
Using Lemma 1, we get: N
T ( ( T - ~ & ~S) C 2-2k-1(2N-2k - 1). k =O
On the other hand N
T(rZN)=
N
2-k(2N-2k+ 1 ) >
C 2-2k-1(2N-2k- 1). k =O
k=l
Consequently, there is no (T satisfying cr r2, = KN,N Let us note that stronger results can be obtained if we apply the technique introduced in this paper to study the following sequence of graphs:
ri= (Vi, Vi+l,. . . , V,,)
where
rl= r=(Vl, V,, . . . , V,,).
282
V.G. Tupitsyn
Another method of approaching the same problem can be found in [l].
References [l] V. Tupitsyn, The problem of the bichromatics of graphs using the language of Markov’s trigonometric problem of moments, J. Combin. Theory (1) (A) (1978).
Annals of Discrete Mathematics 9 (1980) 283 @ North-Holland Publishing Company.
ALGORITHMS FOR DETERMINING THE GENUS OF A GRAPH AND RELATED PROBLEMS I.S. FILOTTI Columbia University, New York, N Y 10027, USA
Abstract There are now algorithms for determining whether a graph G is embeddable in a surface of genus g (Filotti (1978), Filotti and Miller (1979)). Such algorithms run in time O(nmg),where n is the number of vertices of the graph. The algorithms provide an embedding if there is one. In this talk I shall discuss improvements to these algorithms and their relations to other problems in topological graph theory.
283
This Page Intentionally Left Blank
Annals of Discrete Mathematics 9 (1980) 285-290 0 North-Holland Publishing Company
ON THE n-CLIQUE CHROMATIC NUMBER OF COMPLEMENTARY GRAPHS Nirmala ACHUTHAN Star.-Math. Diuision. Indian Statistical Institute. 203. Barrackpore Trunk Road. Calcutta 700035. India
In this paper we discuss the Nordhaus-Gaddum type problem for the n-clique chromatic number defined by Sachs and Schauble. We observe that the lower bounds for x , , ( G ) + x , ( G ) and x,,(G).xn(G)involve the Ramsey numbers.
Introduction and definitions In this paper we consider undirected graphs without loops and multiple edges. A set A of vertices of a graph G = (X, E ) is said to be a clique if (x, y ) E E for all x, y in A. A clique on n vertices is denoted by K,,. For positive integers a and 0, t h e balanced complete 0-partite graph denoted by KE is the graph of order cup whose vertex set is partitioned into p sets Xiof size a each such that x E Xi, y E Xi are adjacent if and only if i f j. A k-coloring of a graph is an assignment of k colors to its vertices so that no two adjacent vertices are assigned the same color. The chromatic number x(G) of a graph G is the smallest integer k for which G has a k-coloring. A k-coloring of a graph gives rise to a partition of the vertex set of the graph into k classes. A partition of the vertex set into x ( G ) classes given by a X(G)-coloring is called a chromatic partition of G. In [4]Sachs and Schauble have given a generalization of the chromatic number. For a graph G = ( X , E ) and an integer n 3 2, a k-coloring by n-cliques is defined to be a partition X , , X,, . . . , X , of X such that n o n-clique is contained entirely in one class of the partition, that is, no n-clique has only one color. Let us call the smallest integer for which such a partition into k-classes exists, the n-clique chromatic number of G and denote it by x,(G). A partition X , , X,, . . . ,Xxn given by a n-clique coloring using xn colors is called a n-clique chromatic partition of G. Nordhaus and Gaddum [3] have proved that if G is a graph of order p 2Jp <x(G)
+ x (G)
p + 1,
p < x ( G ) .x(G)6 (+(P + I)),. Finck [ 11determined the values of x, y for which there exists a graph G of order p with x( G ) = x, x ( G )= y. In this paper we solve the analogous problem for x,,(G). 285
N . Achuthan
286
Given an integer n b 2 , the Ramsey number R(n, n) is the smallest integer k such that every graph of order k contains either a Kn or a En. Throughout this paper we assume that p and n are positive integers and n 3. We shall also use R to denote R(n, n). For definitions not given and notations not explained here, the reader is referred to Harary [2].
Theorem 1. Let G be a graph of order p. Then
where R
= R(n,
n) as stated above.
Proof. Let XI, X,, . . . ,XXcG,be a chromatic partition of G. We write x(G) = m(n - 1)+ m', where 0 =Srn' < n - 1. Define n--1
If rn' # 0 define m' Ym+1=
U Xm(n-l)+j.
j=l
It is easy to see that no Yi contains a n-clique. Hence the partition Y 1 ,Yz, . . . ,Y{x(G)l(n-l)} gives rise to a n-clique coloring of G with {X(G)/(n - 1)) colors. Thus
Similarly it can be proved that
Combining the above two inequalities we get
Now from [3] we have x(G) + x(G) s p + 1. Hence
n-clique chromatic number
287
The upper bound in (2) follows since the geometric mean of xn(G)and xn(G)can never exceed the arithmetic mean. Next we shall prove the inequality
P
xn(G). xn(G)
.
(3)
Let XI, X,,. . . ,&n(G) be a n-clique chromatic partition of G with IX,l = maxi (X,1. Then
First
let
xn(G)= 1. It
suffices to prove
that
xn(G)bp/(R - 1). Let
Y1,Y2, . . . , Y,(c, be a n-clique chromatic partition of d. Since each Y,is &-free
==
in both G and G, we have 1 Yi1 R - 1 for all i and so X , G )
p=
2
~Yi~sxn(G)*(R-l).
i=l
Thus the lower bound in (2) is established in this case. Next let xn(G)b2. We shall prove (3) by induction on p. If p S R - 1 , then (3) is trivial. Assume that (3) holds for all graphs of order < p and let G have order p. Since xn(G)3 2 we have 1X,I < p. Let G, be the subgraph of G induced on X,. Then we have xn(
G) 3 x n (61)
*
By the induction hypothesis
But since GI is &-free this gives
From the inequalities (4), (5) and (6) we get (3) and thus the lower bound in (2) is established. Now the lower bound in (1) follows since ( x ~ ( G+) xn(G))’ 3 4 ~ n ( G*)xn(d)-
This completes the proof of the theorem.
Remark. If p and n are such that {J(4p/(R- 1)))is odd and {J(4p/(R- 1)))’- 1< 4p/(R - l), then any graph G of order p satisfies
for otherwise the lower bound in (2) will be contradicted.
N.Achwthan
288
From the definition of Ramsey numbers it follows that for all t =sR - 1 there exists a graph of order t which we denote -by H[t] such that neither H[t] nor H [ t ] has a n-clique. So x,,(H[t]) = 1= x,,(H[t]). In the following we shall prove that there is a H[R - 11 with a Kn-l. Assume the contrary, no H[R-11 contains a Kn-l or Kn-l. Let H , be an induced subgraph of H[R-l] of order R-n. Then HI has neither K,,-l nor &-l. Let H* = KnPlU H1.Clearly H" is a graph of order R - 1 with no K,, or K,,and with a-K,,-l, a contradiction. Thus there is a graph of order R - 1 which has no K,, or K,, but which has Kn-l. Let us denote this graph by H*[R - 11.
Theorem 2. Let x and y be positive integers satisfying the following inequalities x+ys
p+2n-3 ' n-1
(7)
xey2- P R-1 where R = R(n, n). Then there exists a graph G of order p with x,(G) = x and x n ( G ) = Y.
Proof. Without loss of generality let us assume x 2 y. The theorem is trivial when x = l , so let x 2 2 . By (7) we have p>(x+y-2)(n-1)+1. First let p s ( x + y - l ) ( n - 1 ) - 1 and l s k n - 1 . Then for the graph G = K , U K , write p = ( x + y - 2 ) ( n - l ) + l , where a = ( x - l ) ( n - 1 ) + l and P = ( y - l ) ( n - l ) we have x,,(G)=x and x n ( G )= y and the theorem is proved. Next let pb(x
+ y - l)(n - 1).
(9
Case i: p S ( R - 1)xy - (R - n)(x + y - 1). If y = 1, then p Gx(n - 1). This together with (9) gives p = x(n - 1). Now clearly x , ( K p )= x and xn(Kp)= 1. So let y 2 2 and we shall write p - (n - l)(x
+ y - 1) = (R - 1)r + s,
0 G s < R - 1.
Let A,,, be a (0, 1)-matrix with the following properties. (i) The total number of 1's in A is x+y-l+
{
p-(n - l)(x + y - 1) R-1
(ii) Each entry in the first row and first column is 1. We now construct a graph GI of order
1
p + ( R - n)(x+ y - 1) R-1
n-clique chromatic number
289
where vertices correspond to the 1's of A. Two vertices of G, belonging to the same column of A are joined in GI. No two vertices which belong to the same row are joined and two vertices not in the same row or column may not be joined. We shall now construct a graph G from G I as follows. Replace each vertex u of GI, except possibly one vertex w, by a graph H, where if u belongs to the first column, Kn-1 H,= Kn-l if u belongs to the first row but not the first column, H[R - 11 otherwise. Choose the exceptional vertex w to be outside the first row and first column (this is possible since x 2 y 2 2 ) and replace it by H[R - 13 or H[s] according as s = 0 or not. If u, U' are adjacent in G I then each vertex of H, is joined to every vertex of H,.. If u, u' are not adjacent in G,, then no vertex of H, is joined to any vertex of H,,. Let G be the resulting graph. Then G is of order p and it can be seen that x,(G) = x and x,(G) = y. Case ii: p > ( R - 1)xy - ( R - n)(x + y - 1). In this case let us write p-(R - 1)xy +(R - n)(x+ y - 1) = (R - n ) r + s, 0 6 s < R - n.
From (8) we have r S x + y - l and if s > O then r S x + y - 2 . Consider the graph K : on the vertex set S={(i,j): l ~ i s y l,s j e x } where (i, j) and ( i f , j') are joined if and only if j f j'. Replace the vertex (i, j ) by the graph Gii defined below. If s = 0 H*[R - 11 if i = 1, 1 6 j smin{r, x} or if j = 1, 2 ~ i ~ r - m i n { x}+ r, 1 G.. = or if i 2 2 and j 3 2 , Kn-1 if i = l , m i n { r , x } + l < j s x , En-] if j = 1 , r - m i n { r , x } + 2 s i ~ y . If s > o '
G,
=
H*[R-l]
H[s+n-11 Kn-1
-
*
Kn-1
if i = l , l G j S m i n { r + l , x } - l or if j = 1, 2 ~ i s r + 2 - m i n { r + l , x } or if i 2 2 and j 3 2 , if i = l , j=min{r+l,x}, if i = l , m i n { r + l , x } + l S j < x , if j = I, r + 3 - m i n { r + l , x } s i ~ y .
In the above definition of Gii, we follow the convention that an index set ( 1 : A S 14 p } = fl if A > p. Every vertex of Gii is joined to every vertex of Giti.if j f j' and no vertex of Gii is joined to a vertex of Gi,itif j = j' and if i'. Let G be the resulting graph. It can be seen that x,(G) = x and x,(G) = y. This proves the theorem.
290
N. Achuthan
From the above theorem and remark made after Theorem 1 we get the following:
Theorem 3. All the bounds except the lower bound for x,,(G)+x,,(d) in Theorem 1 are sharp for all p. This lower bound is sharp for all p and n except when p and n are such that {J(4p/(R - 1))) is odd and
When p and n satisfy the above condition we have
and this bound is sharp.
Acknowledgement
I wish to thank Drs. A.R. Rao and S.B. Rao for the discussions I had with them. I thank Dr. S.B. Rao for bringing the problem to my notice.
References [l] H.J.Finck, On the chromatic numbers of a graph and its complement, in: P. Erdos and G. Katona, Eds., Theory of Graphs (Academic Press, London, 1968) 99-113. [2] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [3] E.A. Nordhaus and J.W.Gaddum, On complementary graphs, Am. Math. Monthly 63 (1956) 175-177. [4] H. Sachs and M. Schluble, Uber die Konstruktion von Graphen mit gewissen Firbungseigenschaften, Beitrage Graphentheorie (Teubner, Leipzig, 1968) 131-135.
Annals of Discrete Mathematics 9 (1980) 291-296 @ North-Holland Publishing Company
ON C U ” G
PLANES*
A. SCHRIJVER Department of Mathematics, Eindhouen Uniuersity of Technology, P.O. Box 513, Eindhouen, The Netherlands, and Mathematical Centre, Kruislaan 41 3, Amsterdam, The Netherlands We give a geometrical description of ChvBtal’s version of Gomory’s cutting plane method. Restricting ourselves to rational spaces, we prove that the derived geometrical objects are polyhedra again, and that the method also works for unbounded polyhedra.
1. Introduction For any polyhedron P let PI denote the convex hull of the lattice points contained in P.’ If H is the half-space {x 1 w x s d } , where w is a vector whose components are relatively prime integers and d is a rational number, then one easily sees that HI = {x I wx 4 Ld]}. Geometrically, HI arises by shifting H until its bounding hyperplane contains lattice points. So for half-spaces H there is an easy way to determine HI. Since for each half-space H, the inclusion P = H implies that P r c H I , we know that
where the intersection ranges over all half-spaces H with P c H. We denote this intersection by P’. Below we show that P’ is a polyhedron again, and that P(‘)= PI for some natural number t. (As usual, PC0)= P, and P(‘+l)= P“,’.) This is the essence of Chvhtal’s [l]formalization of Gomory’s [4,5 , 61 cutting plane rnefhod for solving integer linear programming problems (cf. Rosenberg [111). Chvital’s original method applies to bounded polyhedra in real space. However, the fact that the method works for these polyhedra follows from its effectiveness for rational polyhedra (see (i) of Section 4 below). Clearly, we may restrict the range of the intersection (1) to supporting halfspaces, i.e., to half-spaces whose bounding hyperplane supports P. We shall see
* Research supported by the Netherlands organization for the Advancement of Pure Research (Z.W.O.). P is a polyhedron if P = { x I Ax 6 b } for some matrix A and some vector b. When using expressions like A x S b we implicitly assume compatibility of sizes of matrices and vectors. We work within rational spaces, rather than real ones. So any matrix and any vector is supposed to be rational. - A lattice point is an integral vector, i.e., a vector with integer components. wx denotes the standard inner product of vectors w and x. LdJ denotes the lower integer part of a rational number d. For the theory of polyhedra and cones we refer to Grunbaum [7],Rockafellar [lo] and Stoer and Witzgall [13]. 291
292
A. Schrijoer
below that we may restrict the intersection to finitely many supporting halfspaces, namely to those half-spaces corresponding to a so-called totally dual integral system of linear inequalities determining P (cf. (ii) of Section 2 below). Therefore, the use of this method to determine PI depends on the capability to find these half-spaces.
2. Two preliminaries (i) For each polyhedron P the set PI is a polyhedron again. This not surprising fact can be derived from Motzkin’s [9] theorem that each polyhedron P can be decomposed as P = Q + C , where Q is a bounded polyhedron and C is a polyhedral convex cone. Motzkin’s theorem implies also that there are half-spaces H , ,..., H,, L , ,..., Ls such that P I = L , n . . - n L s , P = H , n . . . n H , , and Lic Hifor i = 1 , . . . , s (so the bounding hyperplanes of Liand Hiare parallel). (ii) A system of linear inequalities Ax s b is called totally dual integral if the linear programming minimum
min{yb 1 y 3 0, yA = w}
(2)
is attained by an integral vector y , for each integral vector w for which the minimum exists. Hoffman [8] and Edmonds and Giles [2] showed that if A x S b is totally dual integral and b is integral, then each face of the polyhedron P = {x I A x s b} contains lattice points (i.e., P = PI). That is, if b is integral and the right-hand side of the linear programming duality equation max{wx I Ax s b} = min{yb I y 3 0, yA = w}
(3)
is achieved by an integral vector y for each integral vector w for which the minimum exists, then also the left-hand side is achieved by an integral vector x, for each such vector w. It follows from the results of Giles and Pulleyblank [3] that for each polyhedron P there exists a totally dual integral system Ax s b such that A is integral and P = { x I Ax s b}. (In [12] it is shown that there exists a unique minimal such system, provided that P has nonempty interior, that is, provided that P has full dimension.) ChvBtal’s method to determine PI as described in the present paper calls for algorithmic methods to determine such a totally dual integral system.
3. Theorems Theorem 1. For any polyhedron P the set P’ is a polyhedron again. Proof. Let P = {x I Ax S b}. We may suppose that A is an integral matrix, and, by (ii) of Section 2, that the system A x S b is totally dual integral. We show that P ‘ = { x I A x s Lb]} (where LbJ arises from b by taking componentwise lower integer parts), which yields that P’ is a polyhedron.
On cutting planes
293
First, P'c {x I Ax c Lb]}, as each linear inequality in the system Ax c b gives a half-space H, while the corresponding inequality in Ax s LbJ contains HI. Conversely, suppose H = {x I wx s d} is a half-space containing P as a subset. We may suppose that the components of w are relatively prime integers, so HI = { x I w x s Ld]}. Now darnax{wx IAxCb}=min{yb 1 y 3 0 , y A = w}.
(4)
As the system Ax C b is totally dual integral we know that the minimum in (4) is achieved by some integral vector yo. Therefore LdJ 3 LyobJ 2 yo Lb] ,which yields that wx S Ld] for all x with Ax s LbJ. This implies {x I Ax s L b ] } c H I ' So {x I Ax c L b ] } c P'.
Theorem 2. For any polyhedron P there exists a number t such that P(*)= PI' Proof. We prove the theorem by induction on the dimension of the space, and on the (affine) dimension of (the affine hull of) P. If both are zero the theorem is easy. If the dimension of P is less than the dimension of the space, then P is contained in some hyperplane K. If K contains no lattice points, then it is easy to see that PI = P ' = 8. If K contains lattice points, then there exists an affine transformation of the space which brings K to the subspace KOof vectors with last component zero, and which brings the set of lattice points onto the set of lattice points. Moreover, the image of P is again a polyhedron, say Po. By induction we know that in the space KO we have: (Po)(')= (Po)I,for some natural number t. Since each half-space Ho of KOcan be extended to a half-space H of the original space such that H n K o = H o and H , n K o = ( H 0 ) , , it follows that also in the original space (Po)(')= (Po)I' Since the collection of half-spaces and the set of lattice points are invariant under the affine transformation we know that also p(')= p I' Now suppose the dimension of P is equal to the dimension of the space. Let H = {x I w x G d} be a half-space containing PI as a subset, such that P is contained in some half-space {x I wxcd'}. We prove that there is a number s such that PCs) c H. As PI is the intersection of a finite number of such half-spaces H (cf. (i) of Section 2) the theorem follows. Suppose, to obtain a contradication, that for no s we have that PCs)is contained in H. Since P ' c { x 1 w x s Ld'J}, there exists an integer d " > d such that P ( ' ) c {x I wx
294
A. Schrijoer
Therefore, P(r+u) n K = 8, which implies that P(r+u+l)c {x 1 wx zs d” - l}, contradicting our assumption. We prove that for any polyhedron Q with Q c {x I wx =sd”} one has Q’ n K c (Q n K)’ (and equality follows). It is sufficient to show that for each half-space H with Q n K c H there is a half-space G such that Q c G and GIf l K c H p Let H = { x I u x c e } be such a half-space, where the vector o consists of relatively prime integers. As H I Q n K = Q n { x I-wxC-d”},
(5)
there exist, by Farkas’ lemma, A > 0, and u’, e’ such that = 0’- Aw, e 2 e’- Ad”, and Q c {x u ’ x 6 e’}. We may suppose that A is an integer, since replacing u‘ by u’+ pw, A by A + p, and e’ by e’+ pd”, for some nonnegative number p, does not violate the required properties of u’. Let G = {x I u’x 6 e’}. As u’= u +Aw is integral we have that
I
GI n K c {x I u’x zs Le’] , wx = d“} c {x I ux s [el} = HI,
(6)
which finishes the proof.
4. Some remarks (i) Chvhtal restricted himself to bounded polyhedra, but he considered real spaces. However, the analogue of Theorem 2 for bounded polyhedra in real space may be derived from Theorem 2 as follows (note that the analogue of Theorem 2 is, in general, not true for unbounded polyhedra in real space, as is shown by the polyhedron {(x, x f i ) x 0)). If P is a compact convex region in real euclidean HIwhere the intersection ranges over all rational half-spaces space, define P‘ = H with P c H (a polyhedron is rational if it is determined by rational linear inequalities, i.e., if it is the convex hull of the rational vectors contained in it). Now one easily proves that each compact convex region P is contained in a bounded rational polyhedron Q such that PI = Qp It follows from Theorem 2 that Q(‘)= QI for some t. Since PI c P(‘)c Q(‘), it follows that also P(’)= Pp Chvhtal worked with collections of linear inequalities rather than convex geometrical objects. If 2 is a (possibly infinite) collection of linear inequalities in the vector variable x, then 2” is defined to be the collection of linear inequalities occurring in 2,together with all linear inequalities wx 6 d, where w is an integral vector and d is an integer, such that wx s d’ is a nonnegative linear combination of a finite number of linear inequalities in 9,for some d’< d + 1. Now if P is the set of vectors x satisfying all inequalities in 2,then P’ is the set of vectors x satisfying all inequalities in 2,provided that P is bounded. This follows from the fact that if P c H,where H is a rational half-space, then there exists a rational half-space K such that P c K, HI = KI and the linear inequality defining K is a nonnegative linear combination of a finite number of inequalities
I
nH
On cutting planes
295
occurring in 3’.Hence, for some t, PI is the set of all vectors satisfying all inequalities in 2?(‘). One easily checks that this implies that PI is determined by finitely many inequalities in .@+’). We do not know whether the analogue of Theorem 1 is true in real spaces. We were able to show only that if P is a bounded polyhedron in real space, and P’ has empty intersection with the boundary of P, then P’ is a (rational) polyhedron. (ii) In [12] we proved that for each polyhedron P with full dimension there exists a unique minimal totally dual integral system Ax 6 b, with A integral, such that P = { x I Ax< b}. Now the proof of Theorem 1 above gives: P’= {x I Ax 6 LbJ}.However, not every linear inequality in Ax 6 LbJ is necessary for defining P‘. Otherwise, each face of P of codimension 1 would have a parallel face in PI, which is obviously not true. (iii) As corollaries of Theorem 2 we have: (a) each face of a polyhedron P contains lattice points, if and only if each supporting hyperplane of P contains lattice points (this is equivalent to: P = PI if and only if P = P’; one may derive from this that any totally dual integral system of linear inequalities with integral right-hand sides yields a polyhedron each face of which contains lattice points-cf. (ii) of Section 2); (b) any affine subspace containing no lattice points is contained in a hyperplane containing no lattice points (this is equivalent to: a system Ax = b of linear equations has no integral solution, if and only if there exists a vector y such that yA is integral and yb is not an integer-cf. Van der Waerden [14, Section 1081; for real spaces this theorem may be proved easily by induction on the codimension of the subspace, using cardinality arguments).
References [l] V. ChvBtal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Math. 4 (1973)’305-337. [2] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Ann. Discrete Math. 1 (1977) 185-204. [3] F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Appl. 25 (1979) 191-196. [4] R.E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bull. Am. Math. SOC.64 (1958) 275-278. [5] R.E. Gomory, Solving linear programs in integers, in: R.E. Bellman and M. Hall Jr., Eds., Combinatorial Analysis, Proc. Symp. Appl. Math. X (Am. Math. SOC.,Providence, RI, 1960) 21 1-2 16. [6] R.E. Gomory, An algorithm for integer solutions to linear programs, in: R.L. Graves and P. Wolfe, Eds., Recent Advances in Mathematical Programming (McGraw-Hill, New York, 1963) 269-302. [7] B. Griinbaum, Convex polytopes (Interscience-Wiley, London, 1967). [8] A.J. Hoffman, A generalization of max-flow min-cut, Math. Programming 6 (1974) 352-359. [9] T.S. Motzkin, Beitrage zur Theorie der linearen Ungleichungen, Inaugural Dissertation Basel, Azriel, Jerusalem (1936).
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[lo] R.T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, NJ, 1970). [ll] I. Rosenberg, On Chvital’s cutting plane in integer programming, Math. Operationsforsch. Statist. 6 (1975) 511-522. [12] A. Schrijver, On total dual integrality, Linear Algebra and Appl. to appear. [13] J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions I (Springer, Berlin, 1970). [14] B.L.van der Waerden, Moderne Algebra (Springer, Berlin, 1940).
Annals of Discrete Mathematics 9 (1980) 297-309 @ North-Holland Publishing Company
PROBLEM SESSION
C. BERGE Uniuersiti Pans VI. France
By “path” we mean here an elementary directed path. Conjecture 1. Let G be a directed graph with chromatic number k. Then there exist a path p and an optimal coloring (Sl, S2, . . . , S,) such that ( p n Sil = 1 (i = 1 , 2 , .. . , k ) . Conjecture 2. Let G be a directed graph with stability number k. Then there exist a partition of the vertex-set into disjoint paths (pl, p2, . . . , pk) and an optimal stable set S such that
\ p i n S l = l ( i = 1 , 2,..., k ) .
The two conjectures are true for symmetric graphs, transitive graphs, and tournaments, and generalize well known results of Gallai, Milgram, Roy.
J.-C.BERMOND Uniuersitt? Pans Sud, Orsay, France
Conjecture 1. Let D be an eulerian connected diagraph (that is d + ( x ) = d - ( x ) for every vertex x). Is the cube D3 hamiltonian ( D 3is obtained from D, by joining two vertices x and y by an arc from x to y if and only if there exists in D a directed path of length S 3 from x to y)? Note. This conjecture has been disproved by J. Phcs, with a nice construction of an infinite family of counterexamples.
Problem 2. Let us consider the following graph: the vertices represent the points with integer coordinates of the square [l, n]X[l, n]. Two points are adjacent iff they are “king-connected”, that is (i, j) is adjacent to (i’, j’) if and only if either
i = i’,
j = j’k1
j=j‘,
i=i’&l
i=i‘+1,
j=j’&l
with l S i , i ’ = = n , l S j , j ’ S n . 297
Problem session
298
A subset S of vertices will be called a “chain” if the subgraph generated by S admits a hamiltonian path. The problem is to find the number of subsets (among the 2”’ subsets possibles) which are chains. For example is the number of chains < 2””2?
J.A. BONDY Uniuersity of Waterloo, Ontario, Canada
Let f(n) denote the largest integer k such that every 3-connected 3-regular graph on n vertices contains a cycle of length at least k.
Conjecture (J.A. Bondy, M. Simonovits). There exists a positive constant c such that f ( n ) 2 n c for all n. It is proved in [l] that, for suitable positive constants c1, c2, cC’~df(n)~c,n’og*~og9. Let g(n) denote the largest integer k such that every cyclically 4-edgeconnected, 3-connected 3-regular graph on n vertices contains a cycle of length at least k.
Question. Does there exist a positive constant c such that g ( n ) a c n for all n ? Griinbaum and Malkevitch [2] have shown that the answer to this question is affirmative (with c =): when the graphs under consideration are also assumed to be planar.
References [l] J.A. Bondy and M. Simonovits, Longest cycles in 3-connected 3-regular graphs, Canad. J. Math., to appear. [2] B. Griinbaum and J. Malkevitch, Pairs of edge-disjoint Hamiltonian circuits, Aequationes Math. 14 (1976) 191-196.
M. CHEIN C.N.R.S., Paris, France
M. HABIB Uniuersiti Paris Vl, France
Let G = (X, Y ;E) be a (simple) bipartite graph with vertex sets X and Y, and any edge e E E joining vertices in X to vertices in Y.
299
Problem session
Let M be a matching of G. An M-alternating cycle is a cycle ele2 * that ei E M if i is even and e, E E\ M if i is odd.
*
- e2" such
Problem 1 (Chaty-Chein). Is there a polynomial algorithm for constructing a maximum cardinality matching M in a bipartite graph G so that G has no M-alternating cycle? Problem 2 (Chein-Habib). Characterize those bipartite graphs G = (X, Y; E) such that the family of sets 9l(X)={{XU(MnY)}\(MnX): M is a matching of
G without M-alternating cycles} is the basis set of a matroid on X U Y.
References [l] G. Chaty and M. Chein, Ordered matchings and matchings without alternating cycles in bipartite graphs, Utilitas Math. (to be published). [2] M. Chein and M. Habib, The jump number of dags and posets: an introduction, in this volume. [3] S. Krogdahl, The dependence graph for bases in matroids, Discrete Math. 19 (1977) 47-59.
M. DEZA C.N.R.S., Paris, France
Let d = d , ( l ~ i , j S 6 be ) a metric space on 6 points. We say that it is embeddable in L , if there is an integer k and 6 sets A,, . . . , A, such that \AiA Ail = kdi for l S i , j S 6 (here X A Y denotes the symmetric difference of X and Y, i.e., (X- Y) U (Y - X)). It is easy to show that if d is embeddable in L,, then
2
hihjdijS O
lsi, js6
-
for any A = (A,,. . . ,h6) with all hi integers and A , + * * + h6= 1. Is it true that (1) is also sufficient for d to be embeddable in L,? In fact, is it true that d is embeddable in L , provided only that (1) holds for any permutation of A = (1, 1, 1, -1, -1,O), (2,1,1, -1, -1, -1) and (1,1,1,1, -1, -2)? In the special case that dij = 0 for some i # j , I know that d is embeddable in L , if (1) just holds for any permutation of A = (1,1,1, - 1, - 1,O).
300
Problem session
T.A. DOWLING Ohio Stare Uniuersity, Columbus, U.S.A.
Let S be an mn-set, and let s4 be a family of m-subsets of S with the property that for every partition w = {Bl, B,,. . . ,B,} of S into in blocks of size n, there is an A ~ s such 4 that
I A n B i ( = l , i = l , 2,..., m. Problem. Find f(m, n ) = min ldl.
Known results. (i) f(1, n ) = 1, (ii) f(2, n) = n. The pairs in an optimal family here are the edges of a tree on n + l of the 2n vertices.
(iii) f(m, n ) s n"-'. m-1
Oe = all m-sets containing a fixed point and one from each of m - 1 disjoint n-sets. Conjecture. f(m, n ) = nm-'
J.-G. DUBOIS Uniuersiti du Quebic, Montrial, Canada
On a les formules d'inversion suivantes pour le nombre de partitions en classes de n objets (nombre de Bell &): n
sn = iC= l
Problem session
301
oii S: est le nombre de partitions d’un ensemble de n objets en k classes (nombre de Stirling de deuxibme espbce); n
k=l
oii sk est un nombre de Stirling de premibre espbce ((-l)”-ksk est le nombre de permutations de n symboles ayant k cycles). De fagon analogue, on a les formules d’inversion suivantes pour le nombre de partages de l’entier n : n
nn=2 Pk, k=l
oii Pk est le nombre de partages (non ordonnbs) de l’entier n en k sommants > 0;
c pkn,, n
1=
k=l
oii pk est 1’61Cment (n, k ) de la matrice inverse de celle des nombres Pk. DCbut de la matrice des Pi et des pi: Pi 1 2 3 4 5 6 1 1 0 0 0 0 0 2 .-1 1 0 0 0 0 3 1 ~ 1 1 0 0 4 12- 1 1 0 0 5 1 2 ; 2 1 1 0 6 1 3 : 3 2 1 1 7 1 3 - q 3 2 1 8 1 4 5 9 - 3 2 9 1 4 7 6 : 5 3 10 1 5 8 9 9 - 5
Pi
7 8 9 10 0 0 0 0 0 0 1 1 2 3
0 0 0 0 0 0 0 1 1 2
0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 1 1
1 0 0 0 0 0 0 0 0 2 ---0 0 0 0 0 -1 1 0 0 3 0;-1 1 0 0 0 0 0 0 4 0 0 0 0 1 ;-:I- -1 1 0 0 1 : - 1 -1 1 0 5 0 0 0 6 0 0 1 0 0 1 :--9- -1 -1 1 1 ; 0 -1 -1 1 0 7 -1 0 8 -1 1 0 2 :--o- 0 -1 -1 0 0 -1 0 2 : 0 0 -1 -1 9 1 10 0 -2 1 1 ;--I- 0 0 -1 -1
0 0 0 0 0 0 0 0 0 1
Remarques. (a) Les relations de r6currence pour les Pi sont: k
PA=P:=1,
Pk+k=
2 pn i=l
(Pi = 0 pour i <j ) . (b) On retrouve les valeurs de U,, sur les “marches de l’escalier” de la matrice des Pi. p k = O pour n 3 2 . (c) A remarquer que p:= 1,
Enonce du probleme. Trouver les relations de rkcurrence permettant d’engendrer les ClCments pk de la matrice inverse des nombres PI: de partages de l’entier n en k sommants strictement positifs. Peut-on donner une interpretation B ces nombres pk?
Problem session
302
Two problems in kernel theory P. DUCHET C.N.R.S., Paris, France
A kernel of a directed graph G = (X,U ) is a stable dominating subset K; in other words :
r , ( ~u)K = x.
r,(K) nK = 8,
Conjecture 1 (H. Meyniel, 1976). If every odd circuit of G has at least two chords, then G has a kernel.
-
By a chord of a circuit ul * * up, we mean an arc (vi,vi) with ifi + 1. Particular cases of the conjecture may be found in [l];furthermore, it suffices to prove the conjecture for antisymmetrical graphs (cf. [2])
Conjecture 2. If G has no kernel, then there exists an arc u of G such that G - u has no kernel, unless if G is an odd circuit. This is proved for triangulated graphs.
References [l] P. Duchet, Graphes noyau-parfaits, in this volume. [2] P. Duchet et H. Meyniel, A note on kernel-critical graphs, Discrete Mathematics, to appear.
P. ERDOS Hungarian Academy of Science, Budapest, Hungary
Let f3(n;k, 1) denote the largest number of triangles a graph G on n vertices can have, provided that any induced subgraph of G on k vertices has at most 1 edges. The first nontrivial case is f 3 ( n ; 4 , 4 ) .The question of its value was first raised (in a slightly different form) by Holton at the Canberra meeting in 1977 (see Springer Lecture Notes in Math, No. 682). He assumed that no two triangles have a common edge and asked for the maximum number of triangles. I observed that this is equivalent to the old problem of W. Brown, V.T. S6s and myself: How many triples on n points can one have if no six points span three triangles? Ruzsa and SzemerCdi proved that f3(n; 4 , 4 ) / n 2 + 0 but for every E >0, f3(n;4 , 4 ) / n 2 - E +00. No asymptotic formula for f3(n;4 , 4 ) is known at present. Probably interesting new phenomena will occur for the other values of f3b; k, 0 .
Problem session
303
Several generalizations are possible. For example, we could define f r ( n ;k, 1) with Kr’s playing the role of triangles in the definition of f3(n;k, 1). Also we could generalize the question to both noncomplete graphs as well as to hypergraphs.
P. FRANKL C.N.R.S., Paris, France
Let X,, = {(i, j ) , 1 S i < j d n}. Suppose 9 is a family of (;)-element subsets of X , such that for F, F’ E 9 we have
Conjecture. If n > k > 10, then IS1d (9. Moreover equality holds iff up to isomorphism
9=Po= { F c X I there exists Z c { l , 2, . . . , n}, 121= k, F = { ( i , j ) 1 i < j ~ Z } . I can only prove
R.L. GRAHAM Bell Laboratories, Mumy Hill, U S A .
Question 1. Is there a graph with n vertices and ($+o(l))nlog n edges which contains every possible tree on n vertices as a spanning subtree? Such graphs are known to exist having at most (5/log 4)n log n + O(n) edges. Question 2. How many vertices must a tree T have which contains every tree on n vertices as a subgraph? It is known that the smallest such T satisfies n(l/4+0(1))L~gd o g 2
< v ( ~ ) <2- J. z
nlogn/210g2
n
R.P.GUPTA Ohio State University, Columbus, U.S.A.
Let G be a multigraph with maximum degree A and multiplicity m. Let k be a positive integer and suppose each edge A of G is assigned a set C(A) of k colors.
Problem session
304
Conjecture 1. If G is bipartite and k B A , then there exists a coloration of the edges of G such that (i) each edge A is colored with a color in C(A), and (ii) adjacent edges are colored differently. Conjecture 2. If k 2 A + m, then there exists a coloration of the edges of G such that the conditions (i) and (ii) are satisfied. Note that Konig’s theorem and Vizing’s theorem are special cases of the above conjectures.
Y.O. HAMIDOUNE Uniwrsitk de Pans VI. France
A graph G = (X, E ) is said to be k-critically ..-connected if for all S G X with 0d IS(< k, K(G,-,)= h - IS1 (where K denotes connectivity).
Conjecture. Every noncomplete k-critically h-connected graph has order at most k+h. Remark. Slater conjectured that for k > U h J , is the unique k-critically h-connected graph. I proved that there is no k-critically h-connected graph for k > [ihl with order at least k + h. Hence the above conjecture implies Slater’s conjecture for every k.
Bill JACKSON University of Reading, England
Conjecture. Every 2-connected oriented (simple) graph of minimum in-degree and out-degree at least k, contains a cycle of length at least 2k.
F. JAEGER Laboratoire I.M.A.G., Grenoble, France
T . SWART University of Waterloo, Ontario, Canada
Let us call snark any cyclically-4-edge-connected cubic graph which is not edge-colorable with 3 colors.
Problem session
305
Conjecture 1. There is no snark of girth greater than 6. Conjecture 2 (weaker than Conjecture 1). There is no cyclically-7-edgeconnected snark. M. KLAWE University of Toronto, Ontario, Canada
Does there exist a tree T with dissimilar nodes x and y (i.e., there does not exist any automorphism 4 of T such that +(x) = y) such that T \ {x}= T \ {y} and T \ r ( x ) = T \ r(y)(where T(u)= {w : u adjacent to u})?
J.-M. LABORDE C.N.R.S., Grenoble, France
It has been proved that a simple connected graph G is a hypercube if (1) There exists n E N so that the number of vertices of G is 2" and the number of edges of G is n2"-'; (2) G doesn't contain a triangle; (3) each pair of vertices of G at distance 2 is joined by exactly two paths of length 2; (4)H is not a subgraph of G where H is the graph
Question. Is this also the case if we only assume that (l),(2) and (3) hold, i.e., is condition (4) extraneous? L. LOVASZ and A. HUHN Bolyai Institute, Szeged, Hungary
Consider an inequality
306
Problem session
where cq and p are real numbers such that 1:ai= 0, and PI,.. . ,Pk are lattice polynomials. (1)defines a class of matroids (namely, those matroids in which (1)is identically satisfied in XI, . . . ,X,,). This class is closed under minors.
Question. Can every class of matroids closed under minors be characterized by inequalities of form (l)?
F.S. MULLA Kuwait University, Kuwait
Problem 1. Let G be a finite planar (simple) graph with every vertex of G having degree at least 2. Conjecture 1. There exists a partition of the vertex set of G in four disjoint stable (=independent) sets, two of which are maximal stable sets. Conjecture 2. G has two disjoint maximal stable sets. It is clear that Conj. 1jConj. 2 and Conj. 1 3 the 4-color theorem. Problem 2. Let B be a collection of SQ balls having pi (indistinguishable)balls of type i, 1d i d t. (Thus, p1 + * * * + pt = SQ.) In how many ways can we distribute the balls in B into Q boxes, with each box containing S balls?
C. PAYAN and N.H. XUONG Laboratoire I.M.A.G., Grenoble, France
Let y(G) designate the genus of a (connected) graph G.
Conjecture. If G is a cubic graph, then there exists a vertex x such that (G- X) 2 (G)- 1.
T. RAMSEY University of Hawaii, Honolulu, U.S.A.
Let G be a group of order 2", each element having order 2. Suppose p<< n is a positive integer and E is a subset of G such that for every x E E there is a subgroup H of G satisfying: (i) card(G/H) d 2p; (ii) (x + H) r l E = { x } .
Problem session
307
Conjecture. card(E) = O(nP). I only know that the conjecture holds for p = 1in which case card(E) s n + 1. For p > 1 the problem is completely open.
N. ROBERTSON Ohio State University, Columbus, U.S.A.
Suppose G is a vertex-4-connected graph of girth 4. Is it true that K4,4is the only such graph that does not contain a subdivision of K,?
A. ROSA McMaster University, Hamilton, Ontario, Canada
Let G be a connected undirected graph (without loops or multiple edges), V(G) its vertex set, E(G) its edge set, where (V(G)I23. Let r ( G ) be the automorphism group of G (=permutation group acting on V(G)). For an edge h = {u, u } E E(G), let r h ( G ) be the subgroup of r ( G ) fixing h (and restricted to V(G) \ {u, v } ) . Let Oi(rh(G))denote the orbits of rh(G).A graph G is a A-graph if for every edge h E E(G), I@(I‘,(G))I = 1 for exactly one i (i.e., for every h, rh(G)fixes exactly one vertex). Examples of A-graphs are the odd cycles C2n+l,and the complete bipartite graphs K2,,, with m a 3 .
Problem. Give a simple characterization of A -graphs.
I.G. ROSENBERG Uniuersitt de Monrrtal, Qutbec, Canada
Let G = (V, E ) be a bipartite finite graph with chromatic classes R and B and let F = { R 1 , .. . ,R,} be a family of subsets of R. An F-matching is a subset of E such that: (i) Every r E R is incident to exactly one edge of M, and (ii) for every i = 1,.. . , m the restriction of M to Riconsists of disjoint edges (i.e. for [x‘, y’], [x”, y ” ] M ~ x‘, X ” E R, we have x’ = x ” e y’ = y”).
Example 1. An {R}-matching is just a maximum matching of G.
Problem session
308
Example 2. Let N = { l , 2,..., n}, R = w , B = N x { O } , V = R U B , E = R x B 6.e. G-Kn2,,,)and let F = { { l } + N , { 2 } x N , .. . , { n } x N , N x { l } ,N x { 2 } , . . . , N.x {n}}. The F-matchings agree with the n x n latin squares. The F-matchings may also be described as the maps f from R to B compatible with G and such that all restrictions f 1 Riare injective. What are the conditions for the existence of F-matching for various pairs (G,F)?
S. RUIZ Uniuersidad Catolica de Valparaiso, Chile
Let G be any nontrivial graph. If u is a vertex of G then G, will mean the graph G without the vertex u and its adjacent edges. Is G vertex-transitive if for all vertices u of G, the graphs G, are isomorphic?
P.D. SEYMOUR Merton College, Oxford, England
Let G = (V, E) be an Eulerian graph and let F c E . Suppose that there are circulations & ~ E Fwith , the following properties (a circulation is a flow with no sources or sinks): (i> for f€F,Id+r(f)l= 1 , (ii) for f, f E F, Ic#+fcf')l= 0 unless f = f, (iii) for e E E - F, CfEFI+f(e)l 1. Question. Can these circulations always be chosen all (0, 1)-valued? I conjecture
yes.
J. SPENCER SUNY, Stony Brook, U.S.A.
Call a partition ( X , Y) of the vertex set of the finite graph G a seam if the edges between X and Y are pairwise disjoint. Call G seamless if no nontrivial partition of G is a seam. Question 1. Does there exist a constant K so that whenever X ( G ) = = Kthen , G has a seamless induced subgraph H?
Somewhat stronger is the following. Question 2. Does there exist a constant K so that all critically K-chromatic graphs G are seamless?
309
Problem session
F. STERBOUL Uniuersity de Lille 1, Paris, France
Conjecture. Prove that there exist in Z6p+,-{O}p values xl,
-
. . . ,x p
such that
xp' -x1,. . ., -xp, 2x1,. . ., 2xp, -2x1,. . . , -2xp, x1,
*
*
9
2(x1+ xz), 2(x, -2(x,+x,), x,+
* * *
+ x,+
Xg),
. . . ,2(x, +
+a), . . . , -2(x,+ *+a),
-2(x,+x,+xg),
+xp,
-(x,+
* * *
- -+xp),
are all distinct (where p > 3). This is known to be true for p = 4,5,6,7.
W .D. WALLIS Uniuersity of Newcastle, Australia
Consider an n x n array whose entries are unordered pairs on {1,2,. . . , n} such that: (i) the diagonal entries are (1, l},{2,2}, , . . ,{n, n}; (ii) each symbol appears exactly twice per row and twice per column; (iii) each pair {i, j } , where i # j , appears exactly twice in the array. Such an array is easily constructed from a pair of orthogonal Latin squares of side n: one superimposes the squares, then forgets the ordering on the pairs. However, is it possible to find an array with properties (i)-(iii) which does not come from a pair of orthogonal Latin squares of side n ? I know examples for n = 5 , 6 and 13. Clearly the conditions cannot be satisfied for n = 1,2,3,4. Editorial note. Recently A.F. Mouyart constructed such squares for all n large enough.
J. WOLFMANN Universitt! de Toulon, La Garde, France
Let [F, denote the Galois field, with two elements. If k = 2 t + l is a positive integer, find the minimum cardinality of a set E ~ l F such g that E contains at least half the points of each affine hyperplane of [Fg (viewed as a vector space over IFz). We know the answer if k = 2t. During our talk at this colloquium we gave the only bounds known to us for the case k = 2 t + 1.
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