Graph Theory and Combinatorics, 1988: Proceedings: Cambridge Combinatorial Proceedings in Honour of Paul Erdos

<1

be positive real numbers. If G is an r-uniform hypergraph of order n with every pair of vertices having at most ( p 2+ p)(:It) common neighbours, and with the number of neighbours of every ( r - 1)-set lying in the range { p ( n - r + l ) , p ( n - r + 1) + S}, then G is ( p , a)-jumbled, where 1

2 = - n"-3[p(l - p ) r!

+ p(n - r ) - 10S(p + S / n ) ] .

Proof. Let H be an induced subgraph of G of order k. We shall assume r s k s n - r, otherwise the result is easily checked. Let d = (,! E dH(u),the sum being over all

0E

V(H)('-'). Thus

e ( H ) = -1( k r r-1

) .

For each i, 1 d i =sr, we abbreviate ei(H, G - H ) to ei; note e, = e ( H ) . Denote by

X, the set {aE V(G)('-');la n V ( H ) J= r - 1 - j } . By summing &(o) for u E 4, and using the bounds on d G ( u )given by the conditions of the theorem, we obtain the inequalities

k r-1-J for each j < r. From these inequalites we recover lower and upper bounds bFin and by" for the quantity iei. Substituting ( 1 ) into the inequality with j = 0 gives

by$ = ( r - 1)(

k r-1

) { p ( n - r + 1) - d } S ( r - l)erVl

a ( r - 1)( r - 1 ) { p ( n - r

+ 1) - d + S} =by?'_";.

Pseudo-random hypergraphs

259

These bounds can be substituted into the inequality with j = 1, the results from that being substituted into the inequality with j = 2, and so on. In general, write n - 1 n - 1 -Ik (-l)i(n -r 1) S(j)=( r-1 k-1 i=l k-r+i * Also, setting Aj = {i;1 s i Q j , i p j (mod 2 ) ) and Bj = {i;1 s i sj, i - j (mod 2 ) } , write

+

)( )

and k (n-l)(n-l)-' '(j)=n-r+l r-1 k-1

(n-r+l) iesj k - r + i then it can be verified by induction on j that, for j L 0, b%=(-l)j( r - .1 ) [ p S ( j ) + ( k ) d - ( - l ) J d T ( j ) , I r-1 and r-1 bp!; = (-l)'( I. ) [ p S ( j )+ r - 1 ) d + (-1)j S U ( j ) .

(

'

1 1

In fact we shall not use bj"" in the sequel, but it was required to obtain bpy. We shall assume b y 3 0. This will be the case for values of 8 of practical interest; nevertheless the theorem holds despite this constraint, as shown in [9]. Now, since every pair of vertices in G has at most ( p 2 + + ) ( : : f ) common neighbours, then summing over all pairs of vertices in H, we have

Note that

d H ( a )= (r - j ) e r - j 3 b p ! L 0, so

Multiplying through by two and rearranging as a quadratic in d , we get

J . Havilnnd, A. Thomason

260

It is demonstrated in the Appendix that the following identities hold:

Using these identities and writing

we complete the square for d and obtain

[d - p ( k - r + 1 )

n-1 r-1

+ CdI2

n-2 r-1

This inequality has the form

For real numbers a, b and c with b, c > 0, it can be verified that ( a implies a2s (c + b)2< 2(c2 b2), from which it can be shown that ( a implies a' c 2(c2+ b2) for all a, 6 , c. Thus we have

+

( d -p ( k - r

+ 1))26 2 ( 0 + (c6)2),

+ b)2=zc2 + 6 ) 2s c2

Pseudo-random hypergraphs

261

which we denote in the obvious way as le(H) - p ( : ) I 2 c IH12MM(El+ E2 + E3 + E4 + E5}.

(2)

We now give bounds for S ( j ) , T ( j ) and lXjl, and so for ME,, 1 d i d 5. It is easily checked that IS(i)l c

k(n - r + 1) 1 C ( k - l)r-,-l(n ( r - l)! i = l k

I W l G---

(r - I ) ! ieA;

Therefore,

n2r-3

s 6p-

2r (r - 1)

r! r(r-2)! '

- k)i-l d

k(n - 1)r-l ( r - 2)! '

k(n - 1)1-2 , and 2(r - 2)! (n - k - j)r-l-j(k - r + j + l ) j ( n - k)r-l-jkj-l d k ( r - 2)! ( r - 2)!

( k - l)r-,-l(n - k)i-l d

J. Hauiland, A. Thomason

262

ME5 6 s s

? ( r - 1)'

62(

'c2 ) r-2

k-1

n -r +2 46' r z ( r - 1)'( r - 1 6 2 ( n - 1)2r-4~Zr-2 n - r + 2

(

- i ) y r - 2y4 62 , p - 3 2 2 ~ - 4 (-~ 1) s--

-1

r2(r

n

r!

r ( r -2)!

)-2

.

The proof is completed by substituting these bounds for ME, into (2), and noting that both

+

+

2r-1 (2'-3 T 4 ) ( r - I) 2'(r - 1) and r(r - 2)! r(r - 2)! are less than 10. CI The obvious difference between Proposition A and Theorem 1 is that in the latter we require an upper bound on the ( r - 1)-set degrees for the proof to work. It may be that the dependence of IY on 6 could be removed by a more careful argument in the early stages of the proof. More important, however, is that Theorem 1 never enables us to show that a hypergraph of order n is ( p , O(n(r-')"))-jumbled, the theoretical minimum; the best it affords is ( p , O(n'-t))-jumbled. this is a marked difference from the case r = 2. The remainder of this section provides a test for determining if a hypergraph is jumbled if we know the number of edges in subgraphs of a large fixed order. Hence Theorem 5 generalises Proposition B at the start of this section. To begin with, we prove a technical lemma which will be used heavily later.

Lemma 2. Let rn E N and z E Iw be positive, and suppose xo, . . . ,xr satisfy

C (irn)/Xj= B ( z )

for i = 0, . . . , r.

j=O

Then xi=

(ZY

B(2'z) f o r j = O , . . . ,r.

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263

Proof. Let us fix some value of j , O S j S r , and solve for xi. We need to find numbers yi, 0 =Si S r, such that

in which case we get

and

((i - l)m)j

A. =

1

Now i

di =

A , D ( o ~., . . , (i - l)m, I , (i + l)m,

. . . ,rm)

I=O

provided the At satisfy

= (-~)J+l(+)/j!,SO

This means

’ (-1)’+l I D(Om, . . . , (i - l)m, I , (i + l)m, . . . , rm) yi = -r1 2 (1) D(Om, . . . , (i - l)m, im,(i + l)m, . . . ,rm) I . I=O Oeksr (k#i)

-_ -

j ! i! (r - i)! I=o

Oeksr (k#i)

n (km-imy 1

Osksr (k#i)

264

1. Havihnd, A. Thomason

Next, define S,to be the sum, over all products of t distinct elements from the set {km;0 s k s r, k # i } . Then, expanding the product in the expression for yi gives

Observe it follows at once from (3) that the final sum vanishes if 0 < u <j . Since the sum S,involves (;) terms, each at most mtr!/(r- t ) ! , we may bound lyil by

Finally, returning to Eq. ( 2 ) ,

We are now ready to begin the work leading to the proof of Theorem 5. Lemma 3. Let r a 3 be a positive integer, and let C, n, p and q be positive real numbers with p , q < 1 6 C, such that qn is an integer with 2r s qn < n - 2r. Let G be an r-uniform hypergraph of order n in which for every induced subgraph H of order qn, le(H) -p('':)l < C holds. Then le(H) - p ( t ) l se6rCq-r(l- q)-' for each induced subgraph H of order k.

Proof. Let H be a subgraph of order k 3 qn. If we count the number of edges in each of the ($) subgraphs L of H of order 1 = qn, we get

LcH

LcH

Observe that

(k), kr r - '

I--

;)-r-::(

<-

yr

1--

forl=qn>2r

Pseudo-random hypergraphs

265

Hence

The lemma holds easily for k C 2 r ; now suppose H is a subgraph of order 2r c k c min((1- q)n, qn}. Let F be a subgraph of G - H of order qn, and let L be a subgraph of H of order 1, where 1C 1 C k. Then by the above,

Summing over all ($)subgraphs L for some fixed I , and recalling the definition of ej(H, F), we have

Combining these and dividing by ($) gives

Putting x j = (ej(H, I;) - pN,)/(k),we obtain r

2 (L)?~ = B(erCq-r). j=O

This equation holds for any 1 C k ; selecting r + 1 equations with I = im, 0 6 i 6 r, where rm C k, we derive, via Lemma 2, xi=

(Z,i

B(Te'Cq-').

Choosing m = [ k / r ] ,then m 3 k/2r since k 5 2r, and we have e ( H ) = e,(H, F ) =pNr =p (

=p(

+( k ) A

f)+ f)+

B(22re3rkr(rm)-rCq-r) B(e6rCq-r).

Finally, suppose that (1 - q)n d k 6 qn (this happens only if q 3 i). Summing the number of edges in all subgraphs L of order 1 = (1 - q)n (in a similar manner

J. Havhnd, A. Thomason

266

to the first paragraph of the proof), and using the above we get

and so

Next we extend Lemma 3 by bounding the number of edges in a union of disjoint induced subgraphs of G. The significant feature is that this bound does not depend on the number of subgraphs, as would be the case if we were simply to apply Lemma 3 to each individual subgraph, and then sum.

Lemma 4. Let C, n, p , q and G be as in the statement of Lemma 3, and let s 3 0 be an integer. Let H , , . . . , H, be pairwise disjoint induced subgraphs of G, with orders k l , . . . , k, respectively. Then

Proof. Let A = Ce"q-'(l- q)-'. Since each summand le(Hi) -p(k;)l is, by Lemma 3, bounded by A, we may suppose s 2 e3'/2. Let H be the subgraph of G induced by V ( H i ) ,and let'nj be the number of edges of H meeting exactly j of the sets V ( H i ) , 0 G j G r. If 1 is some integer, 0 d Id s, and we consider all those subgraphs Fof H induced by the union of some 1 of the V ( H i ) , we obtain

u=,

From Lemma 3 we have e ( F ) = p ( ' r ' )+ B(A). Further, if we denote by A!, the number of r-tuples in V(H)(')meeting exactly j of the sets V ( H i ) , we get

)'1;(

i:(s

=j=l

1 -- jj ) y

Hence

As no=No=O we may extend the sum to include j = O , and on writing xj = (nj- ~ N , ) / ( Swe ) ~observe I

C (1)+j

j=O

= B(A).

Pseudo-random hypergraphs

267

If m s s / r is an integer, we derive r + 1 equations by choosing 1 = im, 0 d i s r, and then Lemma 2 yields

(-)jm

2 2

xi =

i

B(2'A).

~~w n1= CG1 e(H,) and Nl = C:=, (ti), so, choosing m = Is/rJ a s ( 1 - r / s ) / r , we have

Ii[e(H,)-p(:)]l

= ~ n ~ - p ~ , ~ = 2e2s s l x ~ l = - - - ~ ( ~ ~ )

m

i=l

2e2r B(2'A) = B(e3'A/2). 0 1- r/s

d-

Theorem 5. Let n, p , a, q, m be positive real numbers with p , q < 1 s a such that qn k an integer with 2r d qn d n - 2r. Let G be an r-uniform hypergraph ( r 3 3 ) of order n in which every induced subgraph H of order qn satisfies le(H) - P ( ~ ; ) I s qna. Then G contains a subgraph G* of order at least (1 - e9'q1-'(1 - q)-'m-')n

which is ( p , ma)-jumbled. Proof. We first construct a hypergraph Go by repeatedly removing 'dense' subgraphs L1,. . . , L, such that e(Li)-p(?) > kiwa, where ILi(= ki and Lj c G - Ui<jLi. We stop when it is no longer possible to choose another L,, and let Li. Let H = ufGlLi and k = IHI = CI=l ki. By Lemma 4, Go = G =

ie(Li)

s

i=l

i=l

p ( i') r

+e3r~/2,

where A = e6rq1-r(l- q)-'na. This gives C;=l kima d e3'A/2 and k s e3'A/2ma. Now construct G* by removing from Go 'sparse' subgraphs Fl, . . . ,F, such that e ( 8 ) - p ( f ; ) < -&ma, where& = 141. By a similar argument, we have IGo- G*l < e3'A/2w(r. Thus JG- G*l < e3'A/wa, as asserted. 0

3. Properties of jumbled hypergraphs We shall now explore some of the consequence of our definition of jumbled hypergraphs. In [12],properties of jumbled graphs, such as the connectivity, the number of hamilton cycles, the number of k-cliques and the contraction number, were estimated. Most the arguments, though sometimes involved, were based upon these next two simple propositions.

I . Haviland, A. T h o r n o n

268

Proposition C ([12]). Let G be a ( p , &)-jumbled graph of order n, and let 0 < E < 1. Then at least ( 1 - E ) n of the vertex degrees of G lie in the range p(n - 1 ) f 1 0 m - l . Froposition D ([12]).Let G be a ( p , a)-jumbled graph of order n, and let O < E < 1. Let H be an induced subgraph of G of order k. Then at least n - Ek of the vertices of G have between p k - 21a.5-l and p k + 2 1 u - l neighbours in H. In this section, we shall first prove versions of these propositions for hypergraphs. For this, the following lemma is required.

Lemma 6. Let G be a ( p , &)-jumbled r-uniform hypergraph, and let S and T be any two vertex-disjoint induced subgraphs of G. Then

where s = IS1 and t = /TI.

Proof. The lemma is clearly true if max{s, t} G 2r, since by definition a k 1. Therefore we assume otherwise, say s k 2r. Let L be a subgraph of S of order I , where 1 < 1 < s. Then

and summing over all such subgraphs L , with 1 fixed, gives

whence

for the number of r-tuples in (S U T)") with exactly j Writing Nj = (;)(.ii) elements in S, we have also

~ ,follows that Putting xi = (ej(S,T ) - ~ N , ) / ( S ) it r

2=o (1),Xj = B(a(1 + t ) ) = B(a(s + t ) ) .

J

If m

QS/T

is an integer, we may obtain r + 1 equations by setting 1 = im,

Pseudo-random hypergraphs

269

0 s i s r, and then Lemma 2 yields xj

(ZY

= - B(Ta(s

+ t)).

Thus, choosing m Islr] 2 s / 2 r since s 2 2r, ej(S, T ) =p 4 + ( s ) ~ x ~

SPY+ 22reZre2'eB(a(s+ t ) ) sp(S)(,

ri) + B(esra(s+ t)).

0

We are now in a position to prove an analogue of Proposition C for hypergraphs.

Lemma 7. Let G be a ( p , a)-jumbled r-uniform hypergraph (r 2 3 ) of order n, with 0 < E < 1. Then at least ( 1 - E ) n of the vertex degrees of G lie in the range p(;:;)

* e6rae-1.

Proof. Let S be a subgraph of order s, and let the sum of the degrees (in G) of the vertices of S be sd. Then r

sd =

2 je,(S, G - S), j=1

and using Lemma 6 we have sd = p

j(S)(" j=1

1

-s)

r-]

+ B(e5'cm i = l

so n-1 d =p( -

+ B(e6'an/B).

Thus taking S to be the [ ~ n / 2 vertices 1 of smallest degree in G, we see that the average of these degrees is at least p(:Z:) - e6raE-'. The proof is completed by taking S to be an [sn/21 vertices of highest degree in G. 0 We also have a version of Proposition D for hypergraphs.

Lemma 8. Let G be a ( p , a)-jumbled r-uniform hypergraph ( r a 3 ) of order n,

J. Haviland, A. Thomason

270

with 0 < E < 1. Let H be an induced subgraph of G of order k. Then at least n - Ek of the vertices of G have behveen p(' k ') - ~ " C Y E - ' and p('! 1) + e7'&ye-' neighbours in H.

Proof. By Lemma 7 applied to the ( p , a)-jumbled hypergraph H , at most ~ k / 3 vertices of H have degrees in H outside the specified range. Let S be a set of s vertices of G - H, and let d be the average degree in H of the vertices in S. Then, by Lemma 6, sd =el($ V ( H ) )=ps(

k r-1

) + B(e5'cu(s + k)).

Hence d =p(' k 1) + B(e5'cu(l + k / s ) ) . Choosing S to be the [ek/31 vertices of G - H of highest degree in H , we see that all but ~ k / vertices 3 of G - H have degree at most p(' k ,) + YE-' in H. A similar argument applied to the vertices of G - H with low degree in H completes the proof. 0 Several of the graph properties studied in [12]have hypergraph analogues. For instance it is easily seen, by a crude estimate, that the clique and independence numbers of a (p,a)-jumbled hypergraph are at most a''('-'), whence the chromatic number is at least na-"('-') . Of more interest is a lower bound on the clique number. For (4, ni)-jumbled graphs, the following proposition from [12], with F = Kk,showed that for k up to about (log, n ) / 2 , the number of k-cliques is approximately that found in a random graph, and so in particular the clique number is at least (log, n)/2.

Proposition E ([12]).Let F be a graph of order r s 3 with m edges, and let z be the order of its automorphism group. Let G be a ( p , &)-jumbled graph of order n, where p S 3. Suppose E sahfies 0 < E < 1 and e'p'n 3 42ar2. Then the number of induced subgraphs of G which are isomorphic to F lies between E)'p'"q(;)-mZ-'n' and ( 1 + E)rpmq(i)-'"Z-*nr, where q = 1 - p .

(1

I

It would be desirable to have a result for hypergraphs in the spirit of Proposition E. A specific reason for doing so, apart from its yielding a lower bound for the clique number, would be to solve this next problem of Erdos and S ~ Sposed , in [7]:

Problem. Let H be an r-uniform hypergraph and f ( n ;H ) be the smallest integer for which every r-uniform hypergraph of n vertices and more than f ( n ;H ) edges contains a subgraph isomorphic to H. An extremal graph belonging to H is a hypergraph G with e ( G ) = f ( l G l ; H ) which does not contain a subgraph isomorphic to H. We define a sequence of hypergraphs Gi (i = 1,2, . . .) to be uniformly distributed if lCil = i, and for every v > 0 there is a c ( q ) , so that for

Pseudo-random hypergraphs

271

every i > io(q) every induced subgraph of Giwith m > qi vertices has (c(r])+ o ( l ) ) ( : ) edges. Is it true that there is no sequence of extremal graphs belonging to H which is uniformly distributed? (In particular, is it true for the case H = K:, the complete 3-uniform hypergraph of order 4?) The proof of Proposition E (with F = K k ) goes roughly as follows. Select a vertex x1 and let Hl be the subgraph spanned by its neighbours. For most choices of xl, IHII =pn. Select a vertex x2 of HI and let H2 be the subgraph spanned by the neighbours of n2 in Hl. Again, for most choices of x2, lH21 =p2n, and so on. In this way ordered k-cliques (xl,. . . ,x k } are counted. To be able to count cliques in a jumbled hypergraph, we would need something to the effect that for each vertex, the (r - 1)-uniform hypergraph induced by its neighbours was jumbled, and that this (r - 1)-uniform hypergraph was in some sense ‘independent’ of the original hypergraph. Such properties will not hold for the general jumbled hypergraph, though even if they do, the analogue of Proposition E may still fail; here is a class of examples. A divkion of the set X = (1, . . . ,n } will be a collection 9 of functions fs, 5 E fl-’),such that fs :(X- 5)+ { -1, 1). The set of all divisions is given the uniform probability distribution, so each division has probability 2-“, m =(n-r+2)(,!!2). Let p and 6 be real numbers, O < p , 6 < 1 . For each f E fl-2)and { x , y } ~ ( X - - ) ( ~ )we define the random variable ~ (y ) k ; f ) = -fs(x)fs(y)6. Thus ~ ( b Y ),; E ) equals 6 iff&) Zfs(y) and equals -6 otherwise. A given division 9 induces a probability distribution on the set of r-uniform hypergraphs with vertex set X as follows: the edges appear independently, and for a E A?’), Pr(a is an edge, given

a=1 -

n

(1 - p ) ” ( ; ) ( l + &(a-f ; 5))= g ( u , 9).

Eed‘-2)

Observe if 6 = 0 this probability equals p . We can think of a as being an edge as a result of at least one success among a set of Bernoulli trials, one for each 6 E a(‘-’), each with probability of failure (1 -p)”(;)(l+ E ( U - f ; f ) ) . (We will assume 6 is sufficiently small that, say, (1 - p ) ” ( ; ) ( l + 6) 6 1). We define the space %,(n,p, S), which is the set of r-uniform hypergraphs with vertex set X, wherein, for a given set A cx’r), Pr(A = E ( G ) )=

c. P4.W n 9F

oeA

g(a,

w.

Hence the probability of generating a given hypergraph G is the expected value of the probability of G given 9.More wieldy expressions for Pr(A c E(G)) are given by the next lemma; prior to stating it, we require some more notation. For A c x’r) and 5 E A?-’), the graph A, has vertex set X and edge set {a - fj; 0 3 5 and a E A}. Further, let T&A) denote the set of eulerian subgraphs of A, (those in which all the vertex degrees are even). Finally, let T ( A ) = T,(A), and for t = r, E T(A) let #t = IUs{a E x’”; u - 5 E E(rE)}J.Thus

n,

n,

212

J. Haviland. A. T h o m o n

#t i s the number of (I in A needed to construct all the eulerian graphs t5c A, which form the components of r.

Lemma 9. Let A c A?). I n the probability space Xr(n,p , a),

Proof. We have

and where E denotes expectation. Define Q(A) to be {(a, 5);o E A, 5 E for each R c Q(A) define R , to be R n (x")x { E } ) for each 5 E X(r--2),and # R to be [{a;(a, 5) E R for some a}l. Then Pr(A c E(G ))= E(

2

n n &(a - 5'; Q )

p'A'-*R(p- l)*R

5 OGRE

RcQ(A)

because the values of E ( * ; 5 ) and E ( * ; 5') are independent if 5 # 5'. Now R, corresponds in an obvious way to a subgraph B, of A,. Given a vertex v of this graph, we may partition the divisions into pairs (9, S'),such that f 5 ( u )= -fi(v) and 4 and 9' otherwise agree. If the degree of u is odd, the value of flofReE ( U - 5; E), given 9,will be minus one times its value under 9'. Hence the final expectation will vanish unless B, is eulerian. On the other hand, if BE is eulerian, then for every f5,

n

- 5; 6) =

&((I

O E R ~

n

UU€E(Be)

-fS(4t&J)6

=

(-WBE),

since the number of edges between fil(-l)and fF'(1) is even. Now, because every RE corresponds to an eulerian subgraph, we see that R, corresponds to t E T ( A ) and # R = #t. Thus

Pseudo-random hypergraphs

273

Finally, on writing a(t)=Ug{ u E ~ )a -; E ~ E ( t e ) } so , that #t= Ia(t)l, we see

o(t)cB

We are now in a position to show that the graphs in Xr(n,p, 6) are most surely jumbled; in fact a considerably stronger statement is true.

Theorem 10. Almost every hypergraph in Xr(n,p , 6 ) has the property that, for each 1,O 6 1 6 r - 3, and for Y EX"),the (r - 1)-uniform hypergraph induced on X

-Y

by the edges containing Y b (p, 0(nr-'-j log n))-jumbled.

Proof. It is sufficient to prove the statement for 1 = r - 3, for then, if Y E X(')and H is a subgraph of the induced subgraph on X - Y , where IHI = k, consider the 3-uniform hypergraph Hp induced on V ( H )- p by Y U p, where p E V(H)(r-3-'). We have

So, let G E Xr(n,p , a), let Y E fi-3), and let G, be the 3-uniform hypergraph induced on X- Y by Y. Let x , y E X - Y and 2 c X - Y - { x , y } . Setting A = {Y U { x , y , z}; z E Z } , we see that for every 5 E fl-2), A, is empty or is a star. Since A, contains no non-empty eulerian subgraphs, Lemma 9 implies Pr({x, y, z} E E(G,);z E 2) = Pr(A c E(G)) =pIA1 =pizI,

so the occurrence of edges of G, containing { x , y} follows a binomial distribution. By standard estimates, the number of edges containing { x , y } lies in the range

I. Haviland, A. Thomason

274

pn f ni logn with probability 1 + O(n-'Ogn),so with probability 1+ O(n2-'"g") every %-tupleof V(G,) is contained in p n f ni log n edges of G,. We now estimate the number of common neighbours of x and y. As the graph Kn-,-l has edge chromatic number at most n - r - 1, the set ( X - v - { x , Y } ) ( ~ ) can be partitioned into sets MI, . . . , Mn-r-l such that for each i and A, p E Mi,A f l p = 0 holds. Moreover [(n - r - 1 ) / 2 ] G lMil s [(n - r - 1)/21. Let W Mi and let A = { ~ U { ~ } U A , ~ U { ~ } U A ; A E W For } . any l j ~ X ( ' - ~ A ) ,, is empty, a path, or a set of independent edges, because the A E W are disjoint. Hence, once again, Pr({x} U A, {y} U A E E(G,); A E W) = Pr(A c E(G,)) =plAl =p2Iw', so the number of A E W with { x } U A and { y } U A in E(G) follows a binomial distribution with probability p 2 . Thus the number of A in Mi with this property is at most p2n/2 + n f log n with probability 1+ O(n-logn),and summing over all Mi we see the number of common neighbours of x and y is at most p2("2') + n4 log n with probability 1 O(n'-'"gn). The same holds for all pairs { x , y } E (X- Y ) ( ~ ) with probability 1 + O(n3-IoK"). Applying Theorem 1 to G,, we see that G, is ( p , U(nf 1ogn))-jumbled with probability 1 + O(n3-'Og"),and this will hold for every v E f l F 3 with ) probability 1 O(n'-'osn)= 1 + o(l), as claimed. 0

+

+

Theorem 10 cannot be extended to I = r - 2. For let v E fl-2), and consider G, and { x , y } ~ ( X - v ) (as ~ )before. If Z c X - v - { x , y } and A = { v U { x } U z , v U { y } U z ; z E Z}, then A, is a complete bipartite graph K,,,,,. From this it follows that IAi

Pr(AcE(G))=piA' j=O

I even

(IAlI. )(-)p -P 1 'J(-6)2J

=+{(p2+ (1 -p)262)'Al+

(p2-

(1 -p)262)'Al}.

Hence the distribution of common neighbours of x and y is bimodal, being the average of two binomial distributions with probabilities p 2 + (1- p ) 2 6 2 and p 2 - (1 - p)262. Certainly the number of common neighbours will almost surely not lie close to p 2 n , so G, will not be ( p , o(n))-jumbled. To some extent, this may explain why the proportion of (r + 1)-cliques in G E X,(n, p , 6 ) is not that found in a random hypergraph, namely pr+', as we proceed to demonstrate.

Theorem 11. Let K c X, lKl= r + 1. Then for G E Xr(n,p,

=p r + l + f

- 2 (

1- p)3a3 + O(b6),

a),

275

Pseudo-random hypergraphs

the last term signifiying p constant and 6 +0. In particular there are values of 6 for which Pr(K(') c G) #p'+l.

Proof. Let BcK"), IBl=j. Then B = { K - { ~ } ; ~ E Y }for some set Y with JYJ=j . Let l j E X(r-'). The graph BE is empty unless l j E K(r-2), in which case p = K - l j E K(3)and BE has one edge for each element of Y n p. So BE contains no non-trivial eulerian subgraphs unless p c Y,when BE is a triangle. There are (4) such p, and hence f , for which this holds. By Lemma 9,

)(p

- 1)'(1-

63)(4). 0

Although we cannot prove that the proportion of ( r + 1)-cliques in a (p, o(n'-'))-jumbled hypergraph is around pr+', it may yet be possible to establish that the number of (r + 1)-cliques in non-zero. This would be enough to answer the above-mentioned question of Erdiis and S6s.

Appendix Here we establish the identities employed in the proof of Theorem 1.

Identity 1.

Proof. We start by demonstrating that the first two expressions are equivalent. Taking the left-hand side, we have (n - l)(n - 1)-lr$l k (k-r+1) r-1 k-1 (r; 1)' -1 r-1

i=l

r-1 i=l

'

-1 r-1

r-1 j

n-r+l k-r+i

'

=(r - 1 )("-') k-r ]el( ) 1 4 1 - 1 h ( - 1 ) i - 1 [ ( k - ~ ~ ~ - l ) + ( k ~ ~ ~ ~ ] i=l

k-r

-r+j

J . Haviland. A. l7wmason

276

To prove that the second two expressions are equivalent, we observe, by expanding binomial coefficients and rearranging,

=

5' (k

r;j

+ 1)(

/=o

n-k-j n-k-r+l

n-r+2 r-1

>=(

1'

the last step following by counting the number of geodesics in ' 2 from (0,O) to (r - 1, n - 2r + 3) passing through ( j , k - r + 2). 0

ldentity 2.

identity 3. r- 1

2 ('

/yo

7

1)2S(j)2IX,[-' - (k - r

= k(k - l)(n - 2 )

r-1

+ k ( nr -- 11

+ 1)2

).

Pseudo-random hypergraphs

277

Proof. Taking the left-hand side of the expression, expanding the binomial coefficients in the first term, and multiplying its numerator and denominator by ( k - r + l ) !(n - k - r + l ) ! ,we obtain

'2'

,:

( k - r + l ) !( n - k - r + l ) !k ! ( n - k ) ( k - r ; j + 1)( n - k - j (n - r)!2 j=O n-k-r+l

(2 - [il x

-'+ '))2 - ( k - r + l ) f rf l ) ( n

(-l)i(n k-r+i

i=l

(k

r ; j + 1)(

j=O

-

(" -r -r1+ 2 ) ( n --k

[i'( k - r ;

j

j=O

:)-'

n - k -j n-k-r+l

r)2}(

+ 1)(

:')(:I

r-1

)(k -

+

n - k -j n - k - r + l )(-')'(k

- k)-l( - ')-' r-1 n-k n-r -r

n-r

+i)- ( k -A)

Note that we have the identity Cl~~((k-rf'+l)(nlkk;il)=( n -rr-+l 2 ) from the proof of identity 2, so making this substitution, our expression becomes

)(n -nk- -r j )((n -nk- -rj

( k * ( k - r ; j + l ) ( n -nk--kr-+j l j=O

(r

n-k

ff

"'(1

-1)

-1

n-r

) - 2 ( - l ) j ( nn- r- )k) ]

-2

( n -k )

1 r - l ( k - r + j + 1) 2( -1)' =k!(n-k)!z . ( n - k ) !( k - r ) ! i=o I ! ( r - j - l ) ! (n - k - j ) ! ( k - j + r ) ! ( k - r +j + 1) 2 . j = o J ! ( r - j - l ) !(n - k - j ) ! ( k - j + r)! r-1

=k! (n - k)!

')

(-1)' ( r : ' ) ( k - r + j + 1 ) J r-1 ( k - r + j + 1) = k ! (n - k ) ! . j = o l !(r - j - l ) !(n - k - j ) ! ( k - j

-2k(k-

r-1

j=o

z

- ( rk- !l ()n! -( kn)-!r ) r-'! z ( r -j 1 ) ( n -nk- -r j -

k!(n-k)! (r-l)!(n-r)!

[z( r-l

+ r)!

)(k-r+j+l)

r-1 n-r-1 )(n-r)+z(r:l)( n-r j )(n-k-j j=o J n-k-j

>1

J. Haviland, A. Thomason

278

k! (n - k)!

- ( r - l)!(n - r ) ! [(n

-.)(,"I): (: I:)I

= k ( k - 1 ) ( n - 2 )+k( n - 1 r-1 r-1

+

).

0

References [l] B. Bollobh, Random Graphs (Academic Press, London, 1985). [2] B. Bollobhs and A. Thomason, Parallel sorting, Discrete App. Math. 6 (1983) 1-11. [3] B. BollobBs and W.F. de la Vega, The diameter of random regular graphs, Combinatorica 2 (1982) 125-134. [4] F.R.K.Chung, On concentrators, superconcentrators, generalisers and non-blocking networks, Bell Syst. Tech. J. 58 (1978) 1765-1777. [5] F.R.K. Chung, R.L. Graham and R.M. Wilson, Quasi-random graphs (preprint). [6] P. E r d h , Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947) 292-294. , Ramsey-TurBn type theorems for hypergraphs, Combinatorica 2 [7] P. Erdiis and V.T.S ~ S On (1982) 289-295. [8] P. Erdos and J. Spencer, Imbalances in k-colorations, Networks 1 (1972) 379-385. [9] J. Haviland, Ph.D. thesis, University of Cambridge (in preparation). [lo] P. Koviki, V.T. S6s and P. TurBn, On a problem of K. Zarankiewin, Colloq. Math. 3 (1954) 50-57. [ll] A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Phil. Soc. 95 (1984) 261-265. [12] A. Thomason, Pseudo-random graphs, in Proceedings of Random Graphs, Poznad 1985, M. Karonski, ed., Annals of Discrete Math. 33 (1987) 307-331. 1131 A. Thomason, Random graphs, strongly-regular graphs and pseudo-random graphs, in Surveys in Combinatorics 1987, C. Whitehead, ed., London Math. Soc. Lecture Note Series 123 (1987) 173-195.

Discrete Mathematics 75 (1989) 279-313 North-Holland

279

BOUQUETs OF GEOMETRIC LATTICES: SOME ALGEBRAIC AND TOPOLOGICAL ASPECTS Monique LAURENT* CNRS, Lamsade, Universilk Paris Dauphine, Place du Markchal de Luttre de Tassigny, 75775 Paris Cedex 16, France

Michel DEZA CNRS, UA 212, UniversitP Park 7, 2, Place Jussieu, 75251 Paris Cea'ex 05,France

Introduction Matroid theory is in the center of Combinatorics, Finite Geometry, Lattice theory and Combinatorial Optimization. During the last decades, extensive search was done in order to find a good degree of generality which still preserves the validity of deep results known for matroids. One of such generalizations is the concept of bouquet of matroids introduced in 1983 by Deza, Frank1 and Laurent and studied in a dozen papers (cf. [7, 11, 14, 171 and references mentioned there). The following matroidal features were extended in a satisfactory way till now: -classical axiomatizations and their equivalence (axiomatizations through flats, independent sets, circuits, rank function, closure operator) (cf. [ l l , 171) -operations and extremal theorems for perfect matroid design case (cf. [11, 12,611 -diagram representation and geometrical aspects (cf. [14, 171) -algorithmic and polyhedral aspects (cf. [8, 91) -orientation (cf. [13]). This paper is a follow-up work in the above series of articles on bouquets and it deals especially with the following features: other operations (contraction, restriction and cuts), strong maps and mapping cylinders, representability , topological aspects and, in particular, shellability of various simplicia1 complexes associated with bouquets and relation with connectivity properties. On the other hand, the starting point of this paper was the important paper of Wachs and Walker 1231. We realized that their principal concepts and results (strong map, mapping cylinder, realization theorem) stated for geometric semilattices could be naturally extended for the broader framework of bouquets. * This work was performed while the author was in CNET, Issy Les Moulineaux, France. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

280

M. hurent, M. Deza

We also give new examples of geometric semilattices; actually, our transversal geometries include all examples of [23]. The paper is organized as follows. Sections 1 to 3 recall briefly generalities on bouquets of matroids: main axiomatizations (through flats in Section 1 and through independent sets and circuits in Section 3), central examples of transversal geometries and d-injection geometries in Section 2, structure of the semilattice 9($) of all bouquets with given independence system 9 in Section 4. In Section 5, we introduce bouquets of geometric lattices as the lattice representation of bouquets of matroids. In Section 6, we consider operations on bouquets: contraction, restriction and cuts and we study their effect on the independence system of the bouquet. In Section 7, we study strong maps on bouquets; we give two new examples of strong maps coming from the closure operator between comparable bouquets having the same independence system (Theorem 7.2) and from the projection map for transversal matroid designs (Theorem 7.6). Then, using the mapping cylinder construction, we prove a realization theorem (Corollary 7.20) which essentially says that every bouquet with M branches can be obtained from a “better” bouquet having only m
1. Flat axioms for bouquets of matroids

We first define bouquets of matroids through their flat axioms which are a direct relaxation of the matroidal axioms.

Dewtion 1.1. Axiomatization through flats. Let X be a finite set and X I , . . . ,X,,, be subsets of X forming a clutter, i.e. X,q! X,for all i # j. Let %, q ,. . . , 9isbe pairwise disjoint families of subsets of X and 3 = $& U * - U gS.Then, the family 3 is called a bouquet of matroids on X of rank s with roofs XI, . . . , X,,, if ( F l ) % ~ U 7 = ~ 2 ~ ~, .a. n. ,dX,,,E%. X~ (F2) 3 is stable under intersection, i.e. G n G’ E % for all G , G‘ E 3. (F3) if G E %i, G’ E and G 5 G’, then i <j (F4) if GE%,.for O S r S s - 1 , ~ E X - Gand G U X E U E ~then ~ ~ there , such that G U x E G‘. exists (a unique) G‘ E Elements of 3 are called flats or closed sets, elements of 3,.are called r-flats or

Bouquets of geometric lattices

281

puts of rank I , for 0 G r G s. The roofs of the geometry $3 are the maximal (for set inclusion) flats. Clearly, for each i E [l,m ] , the interval 4. = Cefl[0, Xi] is the set of flats of a matroid on Xi and Ce= Ad1 U - - - U Ad, is therefore the “bouquet” of the m matroids At,, its rank s being the maximum value of the ranks of the matroids Ati. We will sometimes refer to the matroids 4. composing the bouquet Ce as its brunches or jowers. The above observation yields naturally the following equivalent definition for bouquets which essentially says that a union (in the set of theoretical sense) of matroids is a bouquet if and only if it is stable under intersection.

Definition 1.2. Axiomatization through flats. . . . ,X, be a clutter of subsets of X. A family 93 of Let X be a finite set and X1, subsets of X is the set of flats of a bouquet of matroids on X with roofs XI,. . . ,X, if (Fl) Ceer LJEl 2xi and XI, . . . ,X, E $4 (F2) Ce is stable under intersection (F3’) 4. = Cell[@, Xi] is the set of flats of a matroid on Xifor each i E [l,m]. We recall some more definitions for bouquets. Let Ce be a bouquet of patroids of rank s on X with roofs XI, . . . ,X,,,. When every subset of a roof is a flat, one says that Ce is free. Ce is called weff-cutwhen all roofs have the same rank s, i.e. when the set of roofs coincides with the set Ces of s-flats. Obviously, there is a unique flat of rank 0 and one can assume w.1.o.g. that it is 0. The bouquet Ce is called simpfe when all 1-flats have cardinality 1. When, for each r E [0, s], all r-flats have the same cardinality f,, the bouquet is called a design with parameters ( l o , . . , ,Zs). An epimorphism between two bouquets of matroids Ce, c4‘ is a surjective mapping from Ce onto 93’ which preserves rank and incidence; if, furthermore, it is one-to-one, then it is called an isomorphism. Given a matroid A, the bouquet Ce is called At-unisupported if, for each i E [l,m],the matroids Jac and 4. = Cen [O, Xi]are isomorphic. Clearly, if Ce is unisupported, then Ce is well-cut and all roofs have same cardinality. As we shall see, transversal matroid designs represent an important class of unisupported bouquets.

2. Examples of bouquets: Transversal and injection geometries Bouquets of matroids are, in fact, a special case of the more general concept of $-squashed geometries introduced by Deza and Frank1 in [ l f , 121. In brief, 9 being a clutter of subsets of a finite set X, %-squashed geometries are a generalization of the matroidal structure in which the flats, in addition to satisfymg some axioms similar to axioms (Fl)-(F4) from Definition 1.1, have to be contained in some element of 9; this amounts to replace in Definition 1.1 the clutter of the roofs by the “covering” clutter 9 (i.e. each roof Xiis contained in

‘

M. Laurent, M. Deza

282

some F E 9).By specifying the clutter 9,one obtains various classes of squashed geometries, such as transversal geometries, permutation geometries (6, 71, injection geometries [11] and more generally, d-transversal geometries [14, 171. We recall now precisely the classes of transversal geometries and d-injection geometries that we will especially consider in this paper.

Definition 2.1. Let N , , . . . , Nd be d (d 3 2) finite sets. For (Y E [l,d], a set A, A z N l x . . X Nd, is called injective by N, if, for all distinct elements a = (al, . . . , ad), b = ( b , , . . . , bd) of A, a, # 6, holds. Then, a set A c Nl x . x Nd is called d-injective if A is injective by N, for all (YE [l, d ] and a set A E N,x N2is called transversal if A is injective by N,. One denotes by .T(N,, N2)the family of all transversal subsets of Nl x N2 and . . . , N,,) the family of all d-injective subsets of Nl x . x Nd. by 2(N,,

--

--

Definition 2.2. Let %,, . . . , %, be pairwise disjoint families of subsets of X=N, x N2 (resp. X = Nl x . . x N d ) and %= %,U - .. U 3,. Then 3 is called a transversal geomerry (resp. d-injection geometry) on X of rank s if: (Gl) each set G E 3 is transversal (resp. d-injectif) (G2) 3 is stable under intersection (G3) if G E 3, G' E and G s G', then i < j (G4) if G E 3, for 0 s r s s - 1, x E X- G and G U x is transversal (resp. d-injectif), then there exists (a unique) G' E %r+l such that G U x c G'.

-

When the geometry '3 is a design with parameters (lo, . . . , l,), then 3 is called a transversal matroid design (resp. d-injection design). In this case, one can easily compute the number of r-flats for O G r S s (cf. [12]); in particular, for a transversal matroid design % of rank s on 11, n] x [l,m], one has: J%,l= ms and for a d-injection design 3 of rank s on 11, n,], one has: 13sl= ly=o (n,- l,)/Vs - 1,). As noted in 1121, transversal matroid designs arise as extremal intersecting families of transversal sets; more precisely, if d is a family of transversal subsets of [l,n ] x [l, m] such that JAflA ' JE {11, . . . , f , } for A # A ' E ,rS, then, for n big enough, Id1 s rns and equality holds if and only if ,rS is the set of roofs of a transversal matroid design. This result can be rephrased in coding theory terminology; for this, see that any transversal subset of [ l , n ] X [ l , m ] of cardinality n can be represented as an n-tuple of [l,m]" and thus transversal matroid designs correspond to extremal codes of length n over the alphabet with rn letters and with a prescribed number of distances. Similarly, d-injection designs correspond to extremal intersecting families of d-injective sets. Notice that any d-injective subset of [ l , nId of cardinality n can be written as {(i, a;, . . . ,a:): i E [l,n ] } and thus be viewed as a set (u2,. . . , od) of d - 1 permutations of [ l , n ] with q ( i ) = a; for i E [ l , n ] , j E [2, d]. Hence the set of roofs of a d-injection design on 11, n]" with f ,= n can be seen as a subset of

nP=,

n:'=,

283

Bouquets of geometric lattices

the group (Yn)d-l, Yndenoting the symmetric group of order n. The case when it is a subgroup is particularly interesting and we refer to [7] for the case d = 2 and to [17] for some results in general case. Transversal and d-injection geometries are highly structured objects; so, most of them are unisupported. For this, let pi denote the ith projection from the product set N, X - X Nd onto Ni.Let 3 be a transversal geometry on ZV, x N2 or a d-injection geometry on NIX XNd; then, for each roof Xiof 3, the matroids Ai= %rI[fl, Xi] and p,(&.) are isomorphic and, if 3 is a design, then pl(&.) is a perfect matroid design (PMD) with the same parameters. In some cases, the matroid pl(&) does not depend on the choice of the roof Xi, i.e. the matroids 4. are pairwise isomorphic for all roofs Xi.

--

Proposition 2.3 [16]. Let 3 be a-transversal matroid design on [l,n] x [l,m ] with 1, = n. Then, h!=pl(&.) is a fixed PMD on [l, n] for all roofs Xiof 3 and the projection p , is an epimorphism from % onto A.

nf=,

Proposition 2.4 [17]. Let % be a d-injection design of rank s on [l,nil with 1, = n l . Assume that one of the following conditions holds: (i) 3 is concentrated, i.e. for all G, G' E %, there exists G " E3 such that Pl(G) nP d W =P d G n G'') (ii) n1 = - - * = nd = n and the set of roofs of 3 forms a subgroup of (Yn"n)-'. Then, A = p l ( & . ) is a fixed PMD on [1,n] and the projection p 1 is an epimorphism from 3 onto A. We will see in Section 7 that, in transversal case, the projection p 1 is an example of strong map. We now survey some of the known examples of transversal geometries. Example 2.5. 9(Nl, N2) is a (full) transversal matroid design. Clearly, Y(Nl, N2) can also be defined as the set: L(N,, N2)={ ( A ,f ) : A r N , and f : A + N 2 mapping} (this example is due to Delsarte, [lo]).

Example 2.6. Let V,, V, be finite dimension vector spaces over the finite field GF(q). The set Yv(Vl, V,) = {W S Vl X V2:dim(pl(W)) =dim W} is a (linear) transversal matroid design on V, X V2. It is easy to see that .Tv(V,, V,) is isomorphic to the set: L,(Vl, V2) = {(W, f ) : W S V, and f E Lin(W, V2)} (this example was considered by Stanton [21] who calls it the semilattice of bilinear forms). One defines similarly the affine analogue of the above set. Example 2.7. A transversal matroid design on [l, n] x GF(m) with 1, = n is said to be linear if its set of roofs-when viewed as a subset of GF(m)" -forms a vector subspace. We refer to [7] for many examples of linear transversal matroid designs and for the exposition of a sufficient and necessary condition for their existence (Prop. 4.4 in [7]).

284

M. hurent, M. Deza

Example 2.8. Let E be a set of mappings from [l,n ] to [l, m ] which is sharply t-transitive (i.e. for all distinct elements xl, . . . ,xt E [l, n ] and all elements y,, . . . , yr E [l, m ] , there exists a unique f E E such that f ( x J = y, for i E [l,t]). Then, the meet semilattice generated by the sets {(x,f(x):x E [l,n ] } is a transversal matroid design with parameters (0,1, 2, . . . , t - 1, n) (cf. Prop. 3.8 in (141). Note that sharply t-transitive sets of mappings are well known objects; so they correspond, in fact, to transversal t-designs (from Hanani, [15]), or, equivalently, to orthogonal arrays of strength t (precisely to OA(m, n ; 1) with order rn, index 1 and degree n) and also, for m prime power, to MDS-codes (cf. 1191).

Example 2.9. Let V be a finite dimension vector space over GF(q), A ( V ) denote the family of affine subspaces of V and H c A ( V ) the family of affine hyperplanes. Any affine subspace S can be identified with the set H ( S ) of hyperplanes containing S. If one considers the partition of H into the parallelism classes, then the collection: d ( V ) * = ( H ( S ) :S f 0 and S E A ( V ) } is a transversal matroid design on H (this example is taken from [23] where the poset A( V)* - {0} ordered by the reverse inclusion is considered instead).

There are many examples of d-injection geometries (cf. [ll, 121); let us simply mention that examples 2.6, 2.7, 2.8 have analogues for the injective case and we recall the following:

Example 2.10. $(A',, . . . , &) is a (full) d-injection design.

3. Other axiomatizations for bouquets of matroids It is a well known fact that a matroid can be equivalently defined through the axioms of its flats, independent sets, circuits (or stigmes), rank function, closure operator (cf. [22]). The same holds for bouquets of matroids for which we recall the main axioms that we will need throughout the paper; we refer to [8, 171 for an extensive treatment of various axiomatizations of bouquets. Let 52 be the set of flats of a bouquet of matroids of rank s on X with roofs X,, . . . , X,,, and, for i E [l, m ] , A, = %n[0, X,]be the matroid determined on X,.For each i E 11, m ] , let us denote by r,, a,,9,, Sq the rank function, the closure operator, the family of independent sets, the family of stigmes, respectively, of the matroid 4.. Then, one is naturally led to define the rank function r, the closure operator (7, the family 9 of independent sets, the family 9 of circuits of % as follows: -the family of independent sets is: 9 = 9, U * . . U $, -the family of circuits is the family 9 of all minimal dependent sets, i.e. D E 9 if andonlyif D $ $ a n d D - x ~ 9 f o r a l l x ~ D .

Bouquets of geometric lattices

285

At this point, let us note that the family 9 of independent sets is an independence system (IS, for short) on X, i:e. it satisfies: (10) ifZE$andJEZ, t h e n J E 9 and the family 9is a clutter, i.e. it satisfies: ( D l ) if D,D‘ E 9and D E D’,then D = D’. Furthermore, the family 9can be partitioned into 9= Y U % where Y = 9n 2xt) and % = 9- Y, with g. = 9 II 2xi being the collection of stigmes of 4.. Elements of Y are called stigmes - they correspond to the “matroidal” part of 9 - a n d elements of % are called critical sets-they correspond to the “non matroidal” part of 9. In fact, the IS 9 is completely determined by the clutter 9 of circuits and conversely. Actually, the additional information that the bouquet 3 is providing, is, respectively, the decomposition of 9 as the union of the m matroidal IS: . . . ,9mand the decomposition of 9into the stigmes Y and the critical sets %. The rank function r and the closure operator (T of the bouquet (8 are defined as follows: -for a set A & X i for some i E [l,m ] , r ( A ) = ri(A) and a ( A ) = ui(A) -for a set A $ UEl 2x, r ( A ) = co and a ( A ) = XU co where 03 is an “infinity” point. In other words, one considers the rank and the closure only for sets that are contained in some roof of (B. Note, that, from the flat axioms, the above definition is consistent, i.e. ri(A)= rj(A) and q ( A ) = a,(A) for A c Xin X,. Moreover, for A E U= 2x1, , one has: a ( A ) = A U { x $ A : there exists S E Y such that X E S and S s A U x } and r ( A ) = max(lZ1 : I E 9 and Z E A ) ; i.e. r ( . ) coincides with the rank function of the IS 9 on subsets of roofs. We recall the axioms for circuits and independent sets since we will need them in the remaining of the paper.

(wCl

Definition 3.1. Axiomatization through circuits. A family 9 of subsets of X if the family of circuits of a bouquet of matroids on X if 9 can be partitioned into two subfamilies 9,% satisfying: ( D l ) D Q: D’for all distinct D,D‘E 9 (02) if S, S’ E Y, S Z S ’ and x E S n S’, then there exists D E 9 such that DcSUS’-x ( 0 3 ) if S E Y , C E % and ~ E S ~ then C , there exists C ‘ E% such that C’ & S u c - x . Then the roofs of the bouquet are the maximal subsets of X that do not contain any C E %. Definition 3.2. Axiomatization through independent sets. Given a clutter X I ,. . . ,X,,, of subsets of X, a family 2 of subsets of X is the family of independent sets of a bouquet of matroids on X with roofs X I , . . . , X,,,

hi, Laurent, M. Deza

286

if

(11) = 9 n 2xi is the family of independent sets of a matroid on XI, for all i E [I, m]

u=*

(12) 9 = 9 1 (13) if I E 9, n 9,and x EX, - X,, then 1 U x E 9. Let 9 be an IS on X and SB be its family of buses, i.e. 9 is formed by the maximal sets 1 E 9.Recall that the IS 9 is the family of independent sets of a matroid on X (i.e. is a matroidal IS) if it satisfies the following augmentation uxiom: (14) if I, J E 9 and 111 < IJI, then there exists an element x E J - 1 such that IUX€9

or equivalently, if 9 satisfies the following basis exchange uxiom: (B) for all B, B ' E 9 and X E B - B ' , there exists X ' E B ' - B B - X + X ' E 3.

such that

In application, we recall how to construct bouquets from a matroid ([13], example 3.1). Take a matroid A on X, a clutter XI, . . . , X, of subsets of X such that X , n X, is closed in A; define 9 ' as the family of stigmes of A that are contained in some X I , % as the family of minimal sets that are not contained in any X,and 9 = YU %. Then 9 is the family of circuits of a bouquet of matroids Ce with roofs XI, . . . , X,; one says that Ce is induced from the matroid A. We now mention the related notion of representability for bouquets.

D e W o n 3.3. A bouquet of matroids 93 on X is called representable over the field F if there exists a vector space V over F and a mapping Q, from X to V which preserves the rank, i.e. r(rp(A)) = r(A) for all sets A E X with r ( A ) #or, where r(A) denotes the rank of A in Ce and r ( q ( A ) )the vectorial rank of q ( A ) . For instance, bouquets induced from a vectorial matroid and linear transversal matroid designs are representable. The above definition extends the notion of representability introduced in [ 131 for bouquets induced from matroids and coincides with it when the map Q, is one-to-one. It also covers the definition of representability given in [ll] for injection geometries (we point out an error in the formulation in [ll] in Section 6: in the relation "r(A)=r(cp(A))for all A c X", the condition r ( A ) # was omitted). We finally introduce some definitions concerning bouquets of matroids whose IS have specific matroidal properties.

Deftaition 3.4. Let

be an IS on X and Ce be a bouquet of matroids on X with IS

9.Let p 5 1 be an integer. (i) if the IS 9 is matroidal,

then the bouquet Ce is called a geometric semilattice (see Section 5 for more remarks concerning this terminology) (ii) the IS 9 is said to have the p-intersection property if p is the least integer such that 9 can be written as the intersection of p matroids; in this case, one also says that the bouquet Ce has the p-intersection property

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287

(iii) the IS 9 is said to have the p-union property if p is the least integer such that 2 can be decomposed as the bouquet of m matroidal IS (i.e. as the union of m matroidal IS satisfying axioms (Zl)-(Z3)); in this case, one also says that the bouquet 3 has the m-union property.

Theorem 3.5. Any well-cut (i.e. all roofs have the same rank) transversal geometry k a geometric semilattice. Proof. Let 3 be well-cut transversal geometry on [l, n ] X [l, m ] with IS 8;. We prove that 9 is matroidal by showing that the basis exchange axiom (B) holds. For this, let B, B' be two bases of 8; and (i, x ) be an element of B - B'. We prove that there exists an element ( i f ,x ' ) of B' - B such that the set B" = B - (i, x ) + (i', x ' ) is a base of 8;;since 3 is well-cut, it is enough to verify that B " E ~ ;We . first suppose that i € p l ( B ' ) . Hence B' contains an element ( i , x ' ) with x Z x ' . We prove that the set B" = B - (i, x ) + (i, x ' ) is independent. For this, let F be the (s - 1)-flat of 3 containing B - (i, x ) and G be the s-flat containing B; thus, F 5 G. Then the set F U (i, x ' ) is transversal, i.e. i $ p l ( F ) ; else, there exists an element (i, z ) E F and, since F E: G, (i, z ) and (i, x ) are two elements of the transversal set G which implies that z = x and thus B c F, yielding a contradiction. From axiom (G4), there exists an s-flat G' containing F U (i, x ' ) . Now, if B"$$, there exists a circuit D such that (i, x ' ) E D and D E B" G G'; therefore, D is a stigme and (i, x ' ) belongs to the closure F of B - (i, x ) , yielding a contradiction. We now suppose that i $ p l ( B ' ) . Consider again the (s - 1)-flat F containing B - (i, x ) . For rank considerations, p l ( B ' ) # p l ( F ) ; hence, one can take elements it € p l ( B ' )- p l ( F ) and (it, x ' ) E B'. Then ( i ' , x ' ) $ B ; else, one would have (i, x ) = (i', x ' ) , contradicting the fact that i r$pl(B'). Hence the set F U (i', x ' ) is transversal and thus contained in an s-flat which, similarly as before, implies that the set B" = B - ( i , x ) + ( i f ,x ' ) is independent. 0 Corollary 3.6. The full injection geometry 8;(Nl, . . . ,N d ) has the d'-intersection property, for some d' G d. Proof. For i E [l, d], denote by Si the family of all subsets of Nl x - . . x Nd which are injective by Ni; then is a full transversal geometry. Since 9(Nl, . . . , Nd)= Of='=, % and each of the geometries involved is free, i.e. coincides with the its own IS,one deduces that 8;(Nl, . . . ,Nd) can be written as the intersection of d matroids. 0

z.

Problem 3.7. Is it the case that any well-cut d-injection geometry has the d'-intersection property for some d' < d?

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4. The semilattice T(8;)

Let 8; be an IS on X and 9 be its family of circuits. In general, there exist several bouquets of matroids whose IS is 8; or, equivalently, whose family of circuits is 9; in other words, there exist several ways of decomposing 8; as a union of matroids satisfying the independent set axioms (Il)-(I3). For instance, if 93 denotes the set of bases (maximal independent sets) of 9, then, for B E 93, the family 8;B = {I E 8 :I c 8) is obviously a matroidal IS and 9 = UseB 8;B always provides a decomposition of 9 as a (free) bouquet of matroids.

Example 4.1. Let 8; be the IS on [1,4] whose bases are: 12, 13, 23, 14. Then, U 8;23U 8 ; ~and 9 = 8;(12,13,23) U 8 ; ~are two distinct ways of

9 = bI2U

decomposing 8; as bouquet of matroids. Therefore, we are naturally led to consider the collection T(8;)of all bouquets of matroids on X whose IS is 9.The study of Y(8;)has been initiated in [8]; it was motivated by the fact that “best” bouquets in Y(9) permit to find sharp estimations for the performance of the so-called greedy algorithm applied on 9 for searching maximum weight independent sets. Here, by “best” bouquet, we mean a bouquet composed of as few matroids as possible and, as we will see, they are maximal for some order relation on Ye($’). Note also that saying that the IS 8; has the p-union property amounts to saying that there exists a bouquet in Y(8;) composed of p matroids and all other bouquets are composed of at least p matroids. Any bouquet of matroids % of Y(8;)admits 9 as family of circuits and is characterized by the partition of 9into Y U %; hence, one denotes % by %(Y, %) or simply by qY), Y being the set of stigmes, % the collection of critical sets and (Y, %) satisfying axioms (D2)-(03). We define an order relation on T(8;) as follows: cg(Yl, U,) < %(Y2, Z2) if and only if Yl s Y2or, equivalently, c We state some properties of the poset (Y(8;), <). First, notice that axioms (D2)-(03) are trivially verified for the partition of 9 into Y =0, % = 9, implying that the bouquet %(0, 9)= 8; is the least element of ,Ye($).We consider the following family: %’*= {D E 9:there exists D ’E 9, D ’ f D and x E D r l D’

such that D U D ’ - x

E

2}

(4.2)

and define Y*= 9 - %*. It foilows easily from (02)-(D3) that, for any bouquet % E 2’($), one has the inclusion %* E %. Therefore, if axiom ( 0 3 ) holds for the pair (Y*, %*) (note that axiom ( 0 2 ) is always satisfied), then %(Y*, %*) = %* is the greatest element of Z(8;).More precisely, one has the following result: Theorem 4.3 (cf. [8], Prop. 2.1, 2.3). The poset T(8;)is a meet semilattice and the

Bouquets of geometric lattices

meet of any rwo elements %,(Y1,%,)

%&(Y,, %&)is defined by: $2,

289 A

% = %(Y,n

Y*, %l u %2)

-its least element is %(0, 9)= 8; -its atoms are the bouquets %( { S}, 9- {S})for all S E Y* -2 is atomic, i.e. every element %(Y, %) is the join of atoms:

%(Y,

=

v %({S},

S€Y

9- {S})

-2 is a lattice if and only if %(Y*, %*) E 2(8;), i.e. axiom ( 0 3 ) holds for the (Y*, %*), and, in this case, the greatest element of 2($) is %* = %(Y*, %*).

pair

We now give some classes of IS for which the poset 2(2)is a lattice.

Tbeorem 4.4 (Theorem 2.4, [8]). Suppose that the I S 8; is the family of stable sets of a graph, or, equivalently, that ID1 = 2 for all D E 9.Then, the poset S(8;)is a lattice. When the IS 8; is matroidal with 4 as family of flats, then %* = 0 and axiom ( 0 3 ) obviously holds for the pair (0,9); the bouquet %* = %(0,9) coincides in fact with the matroid A and every bouquet % of 2(8;) is a geometric semilattice.

Theorem 4.5. Suppose that 9 is a matroidal IS with Jdc as family of flats. Then, the poset 2(8;) is a lattice with A as greatest element. Proposition 4.6. Let %= %(Y, %) be a bouquet of 2 matroids of 2(8;) with roofs X,,X, and suppose that 8; is not matroidal. Then, % = %*, i.e. Y =Y*, % = %*

holds. Proof. It is enough to show that % = %* holds. For this, suppose for contradiction that there exists a circuit C E % - %*. It is easy to see that all critical sets are of the form { x , y } with ~ E X , - Xand ~ y € X 2 - X 1 . Thus, we have that C = {x, y } with x, y as above. We prove that r(X, - X 2 )= r(X, - X,) = 1 holds. We can suppose that IX2- X,(3 2. Take z E X2- X1 with z # y , then C' = { x , z } E %. From Definition 4.2 of %*, we deduce that C U C' - x = { y , z} $8; and thus { y , z} E 9,Similarly, for all z' EX, -XI with z' Zy,2, { y , z'} E Y which, together with axiom ( D 2 ) , implies that { z , 2') E 9.This implies therefore that r(X, - X,)= 1 and, similarly, r(X, - X 2 )= 1. If r denotes the rank of 9, one obtains that r(X,) = r(X2)= r and r(X1 nX,) = r - 1. Consequently, for any base B of 9, if B c XI, then (Bf (X, l - X2)(= 1 and the same for index 2. We now show that this implies that 8; is a matroidal IS, yielding therefore a contradiction. For this, we show that the basis exchange axiom (B) holds; i.e. for two distinct bases B, B' of 8; with B E X,, B' G X, and an element x E B - B', there exists an e1ementyEB'-Bsuch that B - x + y ~ 8 ; . WhenxEX1-X2, then B - x s X , ;

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M. Laurent, M. Deza

from the augmentation axiom (14) applied to the independent sets B - x , B’ in matroid $ n 2x2, there exists an element y E B’ - B such that B - x y E 9. When x E XInX,, then x E B n X,- B’ fl XI;by applying again (14) to the independent sets (B - x ) n X2 and B‘ n X1,there exists an element y E B‘ n XI- B n X 2 such that the set B f l X2- x y is independent. Let a denote the Xl 2 = B - u. Since the independent set unique element of B - X,, then B f B - { a ,x } + y is contained in XInX2and a E X1- X,,one deduces from axiom (13) that the set B - x + y is independent.

+

+

Theorem 4.7. Let 9 be an IS with the 2-union property. Then the poset 2($)is a lattice whose greatest element 9P is a bouquet of 2 matroids. Proof. Since 9 has the 2-union property, there exists %EE($) which is a bouquet of 2 matroids. One deduces from Proposition 4.6 that Y?= %* and, from Theorem 4.3, that 2(9)is a lattice. Theorem 4.7 does not extend to the case of IS having the m-union property for m 3 3; we refer to [8] for an example of an IS 9 with the 3-union property for which 2($) is not a lattice. Given a bouquet of matroids % of 2($), let m denote the number of roofs of $3, i.e. the number of matroids composing the bouquet Y?. One may ask which are the “best” bouquets in 2’($), i.e. the bouquets composed by the least possible number of matroids. For instance, the least element $ = %(0, 9) is the “worst” bouquet since it involves as many matroids as the number of bases of 9.On the other hand, when $ is a matroidal IS, then the greatest element of 2(9)is the best possible since it is, in fact, a matroid. The following result shows that, generally, if < $, then $ is better than %,, i.e. is composed by less matroids. Note that, if 4 $, then 6c %, and thus no flat G E contains a critical set of %& and the closure a2(G)is well defined.

“beorem 4.8 (Proposition 2.6, [8]). Let %,, $ be two bouquets of matroids of 2($)whose respective numbers of roofs are m , , m2. ff %, < $ holds in E($), then the closure operator a, of $ induces an epimorphkm from Y?l onto ?& and m26 m holds. The above result can be rephrased as follows: if Y?, < $, then the bouquet ?& is obtained from %, by aggregation of the branches of %l; a branch of ?& with roof X, results from the aggregation of all branches of 92, whose roofs are contained in Xz.The best bouquets, i.e. those having minimum number of branches, are and, when 2(9)is a lattice, then the among the maximal elements of 2($) greatest element 9P of 2’($) is the best bouquet. In fact, Theorem 4.8 can be strengthened; when %, 4 $, the closure operator a, induces an epimorphism from FL(Y?l) onto FX($) where, for a bouquet %, F L ( 9 denotes its chain complex formed by the chains of flats: 6 6 s * . s; F, of %.

-

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291

Proposition 4.9. Let g1< & be two bouquets of Lf(9).Let G, G‘ be fiats of & with G 5 G’ and F be apat of g1such that uz(F)= G . Then there exists apat F’ of % such that uz(F’)= G’ and F 5 F’. Proof. One can assume w.1.o.g. that r(G’)= r ( G ) + 1. Then, G’ = u2(GU x ) for some x E G’ - G . Since G = uz(F),x $ F and, in fact, F’ = ul(F U x ) exists. Else, if F’ does not exist, there exists a critical set C E g1 such that x E C and C r F U x ; since F U x s G ‘ , then C $ Ce, and thus C E Y ; , implying that x E uz(F)= G , which yields a contradiction. Observe now that F’ E G‘ holds; take y E F’ = ul(F U x ) , then there exists S E Yl such that y E S and S r F U x U y , but S E Y2 since Yl r 9,, which implies therefore that y E uz(F U x ) G ’ . In fact, G‘ = u,(F’) holds for rank considerations. 0

=

Theorem 4.10. If g1< $ in Lf($), then the closure operator a, of %$induces an epimorphism from FL( Y)onto FL($) which, to a chain: 4 5 5 F, of pats of ?3, associates the chain: uz(4)5 a,(&) 5 - - - 5 a,(&) of pats of &. The proof follows easily from Proposition 4.9. 0

5. Bouquets of geometric lattices

In this section, we look in more detail at the family of flats of a bouquet of matroids viewed as a poset with inclusion as order relation. For the case of matroids, this is a classical approach. It is well known that the poset of flats of a matroid is a geometric lattice and, more precisely, that finite geometric lattices correspond bijectively to simple matroids. Similarly, bouquets of matroids correspond to what we call bouquets of geometric lattices. Definition 5.1. A poset P is a bouquet of geometric lattices if P is a meet semilattice in which every intervaris a geometric lattice. Proposition 5.2. The poset of pats of a bouquet of matroids is a bouquet of geometric Lattices. The above result can be easily seen to hold. Conversely and similarly to the matroidal case, a simple bouquet of matroids can be derived from every bouquet of geometric lattices. Let P be a bouquet of geometric lattices with maximal elements zl,. . . , z, and X as set of atoms; define Xias the set of atoms under zi,then XI, . . . ,X,,,is a clutter of subsets of X.The following facts can be easily checked: -P has a minimum element 0 -P is ranked with rank r ( . ) , i.e. every unrefinable chain from 0 to x E P has the same length r(x)

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292

--define a set Z c X of atoms to be independent if v I exists and r( v I ) = IZl and let $ ( P ) be the family of independent sets of atoms - j ( P ) = $, U U $,, where 9, = {IE $ ( P ) : v I =z zi}is in fact the collection of independent sets of atoms of the geometric lattice P f l [0, zi] and thus $i is a matroidal IS on Xi -the above decomposition is in fact a bouquet of matroids. For this, it suffices to verify that axiom (13) holds. Take Z E $ ( P ) with v I S z i A zj and an atom x with x s zi but x p zi. Hence, v Z v x 6 zi and v I v x # v I which implies that r( v I v x ) = 111 1 and thus I + x E $ ( P ) , stating axiom (13) -if x, y are distinct atoms such that x v y exists, then one has r(x v y ) = 2. Therefore, the bouquet of matroids %(P) on X with roofs X I , . . . ,X,,, whose IS is $ ( P ) is a simple bouquet of matroids. Hence, we have stated the following:

--

+

Proposition 5.3. There is a bijective correspondence between bouquets of geometric lattices and simple bouquets of matroids. Similarly to what happens in the matroidal case, a bouquet of matroids is not completely specified by the bouquet of geometric lattices determined by its flat family. For instance, the bouquets ‘3, 93’whose flat structure is shown below are distinct bouquets that are associated to the same bouquet of geometric lattices.

bouquet 3‘: 1245

235

Remark 5.4. The class of geometric semilattices which has been studied in [23] coincides with the class of bouquets of geometric lattices P for which $ ( P ) is a matroidal IS. At this point, let us mention that this terminology “geometric semilattice” had been also used by Zaslavsky in [24] for denoting in fact the broader class of bouquets of geometric lattices as defined here. We saw in Theorem 3.5 that all well-cut transversal geometries are geometric semilattices. Actually, it turns out that the examples of geometric semilattices considered in [23] are, in fact, transversal geometries; they correspond to Examples 2.5, 2.6, 2.9.

Bouquets of geometric lattices

293

6. Operations on bouquets of matroids There are many known operations on matroids that preserve, in a way, the matroidal properties. Some of them operate on the lattice of flats of the matroid and, as such, are specifically poset operations; this is the case, for instance, for interval taking, direct product, truncation, etc. Some other ones, as restriction or contraction, are more easily described as operations on the family of independent sets of the matroid. Here, we define poset analogues of these operations for bouquets of matroids and we show which properties of the bouquet and, in particular, which matroidal properties of its independence system are carried out through the operations. Let % be a bouquet of matroids on X of rank s with rank function r ( . ) , closure operator a(.) and IS 8;. Given a subset T of X and an integer k, 0 S k s s, there are several ways for constructing new bouquets from %. We consider the following families: (a) upper interval: [T, 4)= { G E (8: G 2 T} (b) T-deletion: %- T = { G - T : G E % and C 2 T } (c) T-contraction: % - T = { G E ’3 r(G U T) # 03 and r(C U T) = r(G) + r ( T ) } (d) T-restriction: % I T = { G f l T : G E %} (e) k-truncation: 9 = { G E (8: r ( G )6 k } . For the operations of interval, deletion and contraction, we obviously suppose that r( T ) # 03. The families [T, 4) and 59 - T are clearly isomorphic as posets; hence it is enough to study, for instance, the T-deletion operation. Before showing that the above families are all bouquets and studying their IS, we recall some preliminary results. Claim 6.1. Let % be a bouquet of matroids and %’ E % be a lower order ideal of %, i.e. if F E%, G E % ’ and F s C , then F E 97. Then 3’ is a bouquet of matroids.

Proof. We use Definition 1.2 for proving that 3’ is a bouquet. It is clear that %’ is stable under intersection and every interval of %’, being also an interval of 3, is a matroid. 0 Given an IS 8; on X and a subset T of X,the following families are obviously IS : (a) T-contraction: 8; . T = {I E 8;: I UJ E 8; for J maximal subset of X - T with JE.9)

(b) T-restriction: 8; I T = {Z E 8;: Z G T } (c) k-truncation: 8;k = { I E 8;: ( I ( k } It will be clear from the context whether, in notation 8; - T, 8; is considered as IS or as flat family of a (free) bouquet.

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294

Proposition 6.2 ([22], Chapter 4). Let A be the family of flats of a matroid on X with 9 as IS and T be a subset of X . Then, (a) the IS 9 T is a matroidal IS on T (no nice characterization of its flats being known) (b) the T-restriction A T is a matroid on T with IS 9 T (c) the k-truncation A' is a matroid on X with IS 9'".

I

I

In the following we study respectively, restriction, deletion, contraction, truncation and general cuts of bouquets; we analyse what is the effect of each of these operations on a bouquet having specific matroidal properties and, in particular, on geometric semilattices and bouquets with the 2-union property.

Tbeorem 6.3. Let % be a bouquet of matroids on X with IS 9 and T be a subset of X . Then 3 T is a bouquet of matroids on T with I S 9 I T. Furthermore, if % is a bouquet of m matroids, then % I T is a bouquet of m' matroids for some m ' , lsm'sm.

I

Proof. We suppose that % is a bouquet of m matroids with roofs XI, . . . ,X,,, and with IS 9.For i E [l, m ] , 9, = {IE 9: I c Xi}is the IS of the matroid .A( = % n Xi]. We denote by m' the number of maximal elements of the collection ( X , n T , . . . , X,,, n T } ; we can assume that these maximal sets are X , fl T for i E (1, . . . , m ' } . Using Definition 1.2, we prove that T is a bouquet of matroids with roofs Xir l T for i E (1, m']. Observe that axioms (Fl), (F2) hold trivially. For i E [ l ,m ' ] , consider the interval %I T n [B, Xi] = { G C l T : G E .A(.}. This interval is therefore the restriction of the matroid .Ui to the set Xin T; hence, from Proposition 6.2, it is a matroid on Xin T whose IS is 9;I Xin T. Thus. axiom (F3') holds and the IS of % 1 T is given by: Uyi, 9; 1 Xifl T = 9lT. 0

[a,

%I

Corollary 6.4. Let % be a bouquet of matroids on X and T be a subset of X . Zf % has the m-union property, then $ 1 T has the m'-union property for some m ' , 0 < m' S m. In particular, if % is a geometric semilattice, then so is % I T.

Proof. Since % has the m-union property, there exists a bouquet % ' ~ 3 ( 9 ) which is composed by exactly m matroids. Since % 1 T and 94' T have the same IS 9 I T, the proof follows from Theorem 6.3 applied to the bouquet 9 . 0

I

Theorem 6.5. Let % be a bouquet of matroids on X with I S 9 and T be a subset of X of finite rank. Then, 94- T is a bouquet of matroids on X - T with IS 3 . ( X - T). Furthermore, if '3 is a bouquet of m matroids, then %- T is a bouquet of m' mutroids for some m ' , 0 S m'

4 m.

Proof. We suppose that % is a bouquet of m matroids with roofs XI, . . . , X,,,. For i E 11, m ] , JFi = { I E 9:Z c Xi}is the IS of the matroid Ai= 94n [8, X i ] . We

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295

denote by m' the number of roofs containing T, we can suppose that T G X I f l - - - nX,,. One can easily verify that (8 - T is a bouquet with roofs X I - T, . . . ,X,, - T and that its rank function p ( . ) is given by: p ( A )= r(A U T ) - r ( T ) for all A E ull2x1-T. We now prove that the IS of 3 - T is 9. ( X - T) . Take first a set Z which is independent for % - T ; hence, Z c Xi - T with i E [ l ,m ' ] and (Z(= p(Z) = r(Z U T ) - r(T), i.e. r(Z U T ) = IZl+ r(T). By applying the augmentation axiom (14) in matroid .Ui, one can find a set J E T such that Z U J is an independent subset of Z U T of cardinality r(Z U T ) ; therefore, IJJ= r( T), implying that I E 3 * ( X - T ) . Conversely, if Z E 9 - ( X - T ) , let J c T such that Z UJ E 9 and r ( T ) = IJI. Then, Z U J is an independent subset of Z U T of size IZl + r( T), implying that p(Z) = IZl, i.e. Z is an independent set for 3 - T. 0 Corollary 6.6. Let % be a bouquet of matroids on X with IS 9and T be a subset of X of finite rank. Then (i) assume that 2(9)is a lattice and % has the m-union property, then % - T has the m'-union property for some m ', 0 6 m' 6 m (ii) if % is a geometric semilattice, then so is % - T (iii) if % has the 2-union property, then % - T has the 2-union property or is a geometric semilattice.

Proof. The assertions (ii), (iii) follow from (i) and Theorems 4.5, 4.7. We now prove (i). By assumption, the greatest element %* of 2(9)is a bouquet of m matroids. Since %<%*, it follows that the set T has also finite rank in %* and, thus, we can apply Theorem 6.5 to the bouquet %* and deduce that %* - T is a bouquet of rn' matroids for m' =sm. Since %* - T and % - T have the same IS, we deduce that % - T has the m"-union property for some m'' C m' S m. 0 Note that the assertion (ii) of Corollary 6.6 is a restatement of Theroem 4.1 [23].

Theorem 6.7. Let % be a bouquet of matroids on X with IS .$and T be a subset of X of finite rank. Then, (8- T is a bouquet of matroids on X - T with IS 9s.X-T). Proof. We suppose that % is a bouquet of m matroids with roofs X I , . . . ,X, and denotes the IS of the matroid with rank function r(.). For iE [1,m ] , &.= % n [ O , X i ] .We denote by m' the number of roofs containing T and we suppose that T E X In - . nX,,,,. Using Claim 6.1, we prove that 3 - T is a bouquet of matroids on X - T by showing that % - T is a lower ideal of 3. For this, take %E % - T, F E % with F c G ; thus, G U T E Xifor some i E 11, m'], r(G U T ) = r(G)+ r ( T ) and we can suppose w.1.o.g. that r ( G ) = r ( F ) + l . Take X E G - F ; then, we have the relation: r(G) + r ( T )= r(G U T ) = r(F U T U x ) S r(F U T ) + 1 and, since r(G)= r ( F ) 1, we deduce that: r(F) + r ( T ) C r(F U T). In matroid .nC,, the

+

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2%

+

reverse inequality holds, implying therefore the equality: r ( F U T) = r(F) r(T), i.e. F E % * T. Therefore, 3 . T is a bouquet of matroids whose roofs are the maximal flats of Ce- T; one denotes them by Y,, . . . , Y,,..Obviously, the rank function p ( . ) of % . T is given by: p ( A ) = r(A) for all sets A contained in some roof for j E [l, m”]. We verify now that the IS of % T is 9 (X - T). For this, take an independent set I of % T. Hence, I c for some j E [l, m”] and its closure G in 3 . T satisfies r(C) = r(Z) = (I1 and t(G U T) = r ( G ) r(T) = IZl+ r(T). By applying axiom (14), one finds a set J such that J E T , I U J E B ; and I I U J I = r ( G U T ) , i.e. IJI = r(T), implying that I E 9 (X- T). Conversely, take I E 9 (X - T) and let J c T such that I U J E 9,IJI = r(T). Define the closure G of I in $4; then I U J is an independent subset of G U T of size IZl + IJI = r(G) r(T), implying the relation: r(GU T) = r ( C ) + r ( T ) and thus that I is an independent set for 3.T.

-

-

-

+

-

-

+

We observe that T-contraction and T-deletion are two operations that yield distinct bouquets ’3- T and %- T which have the same IS 9.(X- T). Hence, Corollary 6.6 remains valid when replacing 3 - T by T and we do not repeat it; note that the assertion (ii) is then a restatement of Theorem 4.3 [23]. In fact, in the poset Y(9. (X- T)), the bouquet % - T is better than the bouquet % T, i.e. 9 - T < %- T, or, in other words, 3 - T is obtained from %. T by aggregation of its flowers. % a

-

Propition 6.8. Let % be a bouquet of matroids on X with IS 9,closure operator a ( . )and let T be a subset of X of finite rank. Then, 9 . T 6 %- T holds in the poset 9(9- (X- T)). Furthermore, the mapping 0: % - T 4 % - T that, to each pat G E Ce . T associates the frat O(G)= a(G U T) - T of Ce - T, is an epimorphismfrom % . T o n t o % - T . Proof. In order to show that % - T < 3 - T holds, we have to verify that all stigmes of 23. T are stigmes of % - T. Let S be a stigme of 3 . T, i.e. S is a circuit of the IS 9 * (X - T) and S is contained in a flat G of Ce- T. Then, the set a(G U T) - T is a flat of 3 - T containing S, which implies that S is a stigme of Ce - T. We observe that the mapping 8 coincides with the closure operator of the bouquet 3 - T and, therefore, Theorem 4.8 implies that 0 is an epimorphism from % . Tonto 3 - T. Example 6.9. Let Ce be the bouquet of matroids on [1,6] whose flat configuration is shown below; its IS 9 has bases: 123, 124, 134,234,345,346,356,456. For the set T = 34, one defines the T-deletion % - T and the T-contraction $3. T whose flat configurations are shown below. Observe that their common IS is 9 (X- T) with bases: 1, 2, 5, 6 . Observe also that 3 - T is a bouquet of 2 matroids while

-

Bouquets of geometric lattices

297

-

% T is a bouquet of 4 matroids

bouquet Ce: 3456

1234

bouquet % - T : 12

56

bouquet 3 . T:

We now consider the operation of deletion of intervals on bouquets. For a bouquet % and a set T, one defines the family: %- [T, 4)= { G E Ce: G $ T} obtained by deleting the upper interval [T, -.) from %. One can obviously suppose that T is a flat of % and the following holds easily:

Proposition 6.10. Let $3 be a bouquet of matroids on X with IS 9 and closure operator a(.) and let T E 3. Then, the family % - [ T , 4) is a bouquet of matroids whose IS is given by: { I E 9:a(Z) $ T } . An atom (1-flat) T of % is called universal if T is contained in all roofs of 3. Note that, for T E $3, one has always the inclusion: % * T s 3 - [T, -.) and that equality holds if and only if T is a universal atom of 3. Hence, if T is a universal atom of 9, then %- [T, 4) has an IS the family 3 - (X - T); therefore, and the Corollary 6.6 remains valid when replacing %- T by 9- [T, 4) assertion (ii) then implies Corollary 4.5 [23]. For matroids, one has the following result:

Proposition 6.11 (Corollary 4.7, [23]). Let A be a matroid and T be a frat of A, then A - [T, 4)is a geometric semilattice. More generally, one can delete several intervals from a bouquet $; so, if T,, . . . , Tp are distinct flats of %, then the family %-U=, [T, 4) is still a bouquet of matroids, also called wounded bouquet. Particularly interesting is the study of wounded matroids. So, we saw above that, when deleting one interval

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from a matroid, one obtains a geometric semilattice. A beautiful result from [23] shows that, conversely, any geometric semilattice can be realized as a matroid with one less interval. We will see in Section 7 that, more generally, any bouquet with the m-union property can be realized as a bouquet of m matroids with one less interval (under the condition that 2(8;)be a lattice). When one deletes several intervals from a matroid, one has the following result:

Propsiton 6.12. Let A be a matroid and T,, . . . , Tp be p distinct fiats of 4. Then, the family A - I&, [T,, 4) is a bouquet of matroids having the p'-intersection property for some p ' p . Proof. Proposition 6.10 implies that A' = A [T,, 4)is a bouquet with IS: 8' = { I E 8;: a(Z) $I T, for i = 1, . . . ,p}; hence 8;' = o f = ,{ I E 8;: a(Z) $ T } , each of the families in the latter intersection being a matroidal IS from Proposition 6.11. 0 The following question is of interest, at least for small values of p , for instance p =2. ProMem 6.13. If % is a bouquet of matroids having the p-intersection property, can % be realized as a matroid with p deleted intervals?

We conclude this section by mentioning the related operation of cuts on bouquets of matroids. Following [13], an elementary cut consists of deleting exactly one roof from the bouquet and a cut is any sequence of elementary cuts. A cut is uniform when it consists of removing all roofs at once. For instance, deletion of intervals is a particular cut and iterated uniform cuts produce the truncation of bouquets.

Proposition 6.14. Let 3 be a bouquet of matroids on X of rank s with IS 9.For k , 0 s k s s, the k-truncation @ is a bouquet of rank k on X with IS d k .

Proof. Easy. 0 ComHary 6.15 (Prop. 4.2, [23]). Zf 3 is a geometric semilattice, then so is any truncation of %.

Proof. It follows from Proposition 6.2, 6.14. 0

7. Strong maps and mapping cylinder operation

The notion of strong map on a geometric lattice is an important tool in matroid theory. It can be extended to bouquets of geometric lattices and, actually, the

Bouquets of geometric lattices

299

definition of strong map adopted by Wachs-Walker ([23]) for geometric semilattices turns out to be well adapted for the general class of bouquets, so we will consider the same notion. We refer to [18] for a further study of strong maps on bouquets.

Definition 7.1. Let Pl, P2 be two bouquets of geometric lattices. A function f : Pl 4 P2 is called a strong map if: (Sl) f is rank reducing, i.e. r2(f ( x ) ) S rl(x) for x E PI (S2) for each atom a E PI and x E Pl, if a v x exists in PI, then f ( a ) v f ( x ) exists in P2 and f ( a ) v f ( x ) =f (a v x ) (S3) for each atom a E Pl and x E Pl, iff ( a ) v f(x) exists in Pz and f ( a ) =jkf ( x ) , then a v x exists in Pl. Note that this definition reduces to the usual definition of strong maps on geometric lattices when Pl, P2 are geometric lattices ([22], chap. 17). It can be verified that a strong map is order preserving, i.e. if y covers x in Pl, then f ( y ) covers or is equal to f ( x ) in P2. Also, (S2) remains valid if one replaces the atom a by any element y E Pl. Most examples of strong maps on geometric lattices from [22] and all examples of strong maps on geometric semilattices from [23] extend easily to the case of general bouquets; we do not repeat them. We introduce two new examples of strong maps: the first one is coming from the closure operator between two comparable bouquets of d;p($), the second one from the projection map for transversal matroid designs.

Theorem 7.2. Let $ be an IS on X and g1,Y& be two bouquets of mafroids of 2($)such that %l < Y&. Then, the map from %l onto % induced by the closure operator u2of % is a surjective rank preserving strong map. Proof. We know from Theorem 4.8 that u, induces an epimorphism from g1 onto ‘92, i.e. a surjective and rank preserving map. We show that u2 is a strong map, i.e. satisfies (S2),(S3). Take a 1-flat F = ul(a) of 3 and G E such that u,(F U G)= H E %l exists; we can suppose that a 4 G, else (S2) trivially holds. We have that u2(F)U u2(G)E u2(H); if a 4 u2(G), then, for rank consideratU u2(G)),which states (S2). Suppose for contradiction ions, u2(H)= u2(u2(F) that a E u2(G);then u2(G)= u2(GU a ) which, since u2is a rank preserving map from onto &, implies that rl(G) = rl(G U a ) , yielding a contradiction with the fact that a $ G . We now verify that (S3)holds. Take a 1-flat F = ul(a) of 3, G E 3 such that u2(F)$ u2(G) and a2(u2(F) U u2(G)) exists. Suppose for contradiction that ol(F U G) does not exist, hence ul(G U a ) does not exist. Thus, there exists a critical set C of 3 such that a E C and C e G U a. Then C is also contained in the flat u2(u2(F) U u2(G))which implies that C is a stigme of &. We therefore deduce that a E u2(G),yielding a contradiction with the fact that u2(F)$uz(G). 0

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300

In application of Theorem 7.2, we have the following examples of surjective rank preserving strong maps; they correspond, for the case of geometric semilattices, to examples 4, 5 , 6 from [23].

Example 7.3. Given an IS $ and a bouquet '3 E dip($) the function:

with closure operator a(.),

Ce

$4

I - a(1).

Example 7.4. Given a matroidal IS 9, A the associated matroid with closure operator a(.) and a bouquet 94 E 2($),the function: 94- A G- a ( G ) . Example 7.5. Given a bouquet of matroids Ce with closure operator a(.) and T a set of finite rank, the function: '3- T-+ '3- T G - u(G U T ) - T. Recall that, if 94 is a transversal matroid design on [l, n ] X [l, m ] with parameters (lo, . . . , Is-l, 1, = n ) , then '3 is A-unisupported where A is a PMD on 11, n ] with the same parameters, i.e. pl(.Ui)= A4 with Ai = %rl[0, Xi]for all roofs Xiof 3, p 1 denoting the first projection from [l, m ] X [l, n ] onto [ l , n ] . Theorem 7.6. Let Ce be a transversal matroid design on [ l , n ] x [ l , n ] with 1, = n and A be its PMD support. Then the map induced from Ce onto A by the

projection p l is a surjective rank preserving strong map.

Proof. We already know from Proposition 2.3 that p 1 induces an epimorphism from Ce onto A.We show that it is a strong map. It is easy to see that axiom (S2) holds. We now verify (S3). Consider a 1-flat F E 92, G E Ce such that p , ( F ) $ p l ( G ) . Since p l ( F ) has rank 1, one deduces that p l ( F )n p I ( G )= 0 and thus that F U G is a transversal set. Therefore, from axiom (G4), there exists a flat G' E '3 such that G U F E G ' , which states (S3). 0

Remark 7.7. This result does not extend to unisupported d-injection designs for d 2 2 , i.e. the projection p 1 is not a strong map from % onto A = p l ( % ) . The reason for this being that, for F, G E '3, the condition p l ( F ) npl(G) = 0 does not imply that F U G is a d-injective set and thus (S3) does not hold. In the following, we show how to relate strong maps between bouquets of matroids to strong maps between their IS or other related bouquets. We first state a preliminary result.

ctaira 7.8. Let Cei be a simple bouquet of matroids on Xiwith IS and family of circuits gi,for i = 1,2. Suppose that 9, has the m,-union property, for i = 1 , 2 , $j

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301

and let f : & be a rank preserving strong map. Given a subset A = { a l , . . . , a,} of X1,we denote by f { A } the subset { f ( a l ) ,. . . ,f ( a , ) } . The following assertions hold: if and only iff { A }E B2 and If {A}( = t (i) A E (ii) if A E g1then, either f ( A )E g2 and If { A } J= t ; conversely, if f { A }E B2 and If {A}I = t, then A E B1 (iii) Suppose that f is surjective and let %,: C: be the families defined by relation (4.2); iff {A} E C: and If {A}I = t, then A E %: (iv) iff is surjective, then m1S m2 holds.

Proof. (i), (ii) are easy to verify. We state (iii). Iff { A }E C:, then, by definition of C:, there exists a set D E g2,D f f { A } , an element x E D f l f { A } such that f {A} U D - x E $$. Since f is surjective, we can find B E g1such that D =f { B } , x = f ( a ) with a E A fl B and f { A f Bl } =f { A }f l f { B } ;therefore A U B - a from (i), thus implying that A E %.: For proving (iv), consider a decomposition of $2 as a bouquet of matroids with m2 roofs: K , . . . , Ym2and define the sets: Zi= { x E XI:f ( x )E for 1G i S m,; then it is a routine to verify that can be decomposed as bouquet of m2 matroids with roofs the sets Z,’s, therefore implying: ml G m2. 0

x}

Theorem 7.9. Let %* be a simple bouquet of matroids with IS $iand closure operator ui, for i = 1, 2, and f : 3 1 4 9& be a rank preserving strong map. Then, there exists a unique strong map f:$14,$’2 such that the diagram below commutes. Furthermore, i f f is surjective, then so is f.

%A% 0 1 1

$1

1 0 2

7 - 9 2

Proof. It can be easily verified, using Claim 7.8(i), that the map f defined by: f(Z) = {f ( a l ) , . . . ,f (a,)} for Z = { a 1 , . . . , a,} E is the unique strong map satisfying Theorem 7.9. 0

Theorem 7.10. Let 9Ii be a simple bouquet of matroids with IS $iand assume that a(,$) is a lattice with greatest element %: whose closure operator is denoted by a : , for i = 1, 2. Let f : 4$ be a surjective rank preserving strong map. Then, there exists a unique map f * : 93: 4 ‘9: such that the diagram below commutes; furthermore f * is rank preserving and surjective strong map.

%L% oil

3:

73;

Proof. We denote by githe circuit family of $i, by Y: (resp. 9’:)the stigmes of (resp. 92:) and by qi(resp. %): the critical sets of %i(resp. 93:). Since is

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M. l a r e n t , M. Deza

a lattice, the family %: is given by relation (4.2), for i = 1, 2. We first recall two relations that we will use in the proof for all I E A,u:(I) = u:(ui(I)for i = 1 , 2 and and for I E ~ f(ul(Z))=u2(f{Z}). ~ , Let F* E be a basis of F*; necessarily, the map f * must satisfy: f * ( F * )=f *(u;(ul(I))) = az(f(al(Z)))= a:(u2(f {I}))= uz(f{Z}). Hence, we are led to define f * by: f * ( F * )= u:(f {I}) where Z is a basis of F* E g.We first verify that f * is well defined, i.e. if u:(I)= u;(I‘), then u:(f{Z}) = uz(f{Z’}). It is enough to prove that f ( a ) E u:(f {1’}) for all a E I. If a E I, then there exists D E 9’;such that a E D c I’ U a ; hence f ( a ) ef{D} c_ f { I f } U f (a). From Claim 7.8(ii), we deduce that f {D} E &. Also, from Claim 7.8(ui), we have that f { D } E SP;, implying that f ( a ) E 4 f {I’}). Obviously f * is rank preserving surjective and we leave it to the reader to verify that f * is a strong map. 0 In the case when 9&,‘92 are geometric semilattices, then Theorem 7.10 remains valid without the assumption that f be surjective and rank preserving, as stated in [23] (Theorem 5.1); actually, a slight modification of our proof also shows it. Cordbuy 7.11. With the notations of Theorem 7.10, suppose that union property for i = 1 , 2 , then m , = m2 holds.

9;.has the mi

proof. From Theorem 7.10, f * is a surjective rank preserving map from the bouquet ’9: on the bouquet g, hence f * maps the m , roofs of %; onto the m2 roofs of g and thus m2d m , holds. The reverse inequality follows from Claim 7.8 (iv), hence implying that m l = m 2 . 0

We now present a poset operation on bouquets of geometric lattices that uses strong maps as essential tool; this is the operation of mapping cylinder which has been introduced in [23] for geometric semilattices. Again it turns out that bouquets of geometric lattices seem to offer the correct level of generality at which the mapping cylinder construction applies nicely. Definition 7.12. Let Pl, P2 be two bouquets of geometric lattices and f :Pl P2 be a strong map. The mapping cylinder C ( P l , P2,f) is the poset whose element set is P, U P2 and whose order relation <=is defined as follows: for x , y E PI U Pz, x <=y if one of the following holds: (i) x < y in Pl when x , y E Pl (ii) x < y in P2 when x, y E P2 (iii) f ( x ) y when x E Pl, y E P2. --I,

Theorem 7.13. Let Pl, P2 be two bouquets of geometric lattices and f :Pl -D P2 be a surjective rank preseruing strong map. Then, the mapping cylinder C(P,, P2,f ) is a bouquet of geomemk lattices.

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303

Remark 7.14. This theorem is a companion to Theorem 6.1 from [23] which states that, when Pl, Pz are geometric semilattices, then C(P,, P 2 , f ) is a geometric semilattice. If one looks carefully at the proof of Theorem 6.1 ([23]), one can notice that, in the first part of it, it is shown that C(Pl, P2,f ) is a bouquet of geometric lattices, using only the assumption that Pl, P2 are bouquets; this part therefore includes the proof of Theorem 7.13 and we do not repeat it. In the last part of the proof of Theorem 6.1 ([23]), using the additional information that the bouquets P l , Pz are geometric semilattices, it is deduced that the bouquet C(P,, P 2 , f ) too is a geometric semilattice; this result will also follow from the more general statement in Corollary 7.16. Theorem 7.15. With the notations of Theorem 7.10, the map q, q:C( %,, g2,f)4 C(%:, g;, f*) defined by q(F) = ar(F) for G E 3,i = 1,2 is a surjective rank preserving strong map.

Proof. q is obviously surjective and rank preserving. From Corollary 7.11 and the proof of Claim 7.8(iv), if 3; has roofs Yl, . . . , Y,, then 93: has for roofs the sets Zi = { a E Xi : f ( a )E Y;.} for i = 1, . . . ,m ; this implies the relations: (a) if S E 9’:,then either S = {a, b} with f ( a ) = f ( b ) , or f{S} E 9; (b) if f{S} E 9’;and If{S}l= IS(,then S E 9’:. Set P = C(%, Y2,f ) and P* = C ( q , ‘;B,*, f*). We show that q is a strong map. We first prove that (S2) holds. For this, take an atom F E P, G E P such that F # G and F v G exists in P. If F=BE%$ and G E g1,then F v G = f ( G ) and thus q ( f ( G ) )= f * ( q ( G ) ) dominates q(F), q ( G ) and, in fact, f * ( q ( G ) )= q(F) v q(G). Now suppose (the other cases are easy) that F = al(a), G = al(Z)E %, and F v G exists in ?&., i.e. al(ZU a ) does not exist; then F v G = ( 7 2 ( f { 0 Uf(a)) and cp(F v G) = a,*(f{lI Uf(a)) dominates f*(cp(F)),f * ( v ( G ) ) and thus q ( F ) , q(G), implying that q ( F ) v q ( G )s f * ( q ( F ) ) v f * ( q ( G )6 ) q ( F v G). Equality holds for rank considerations, after noticing that q ( F ) # q ( G ) ; else, a E a:(Z) which, from (a), implies that f ( a ) E a;(f{Z}), contradicting the fact that F $ G. We now prove that (S3) holds. For this, take an atom F E P, G E P such that q(F) v q ( G ) exists in P* and q(F) # q(G). When q(F) v q ( G ) E %,: then F = ul(a), G = al(Z) and Z U a E dp,, so F v G exists in g1. Suppose now that q(F) v q ( G )E 3;. If F = 0 E ?&., G E g1then f(G) dominates F, G and F v G exists. If F = al(a) E ‘91, G = a2(Z)E ?&.,then q(F) v q ( G ) = a;(Z Uf(a)), implying that Z U f ( a ) E 92;hence a2(ZU f ( a ) ) dominates F, G and F v G exists. Suppose now that F = a l ( a ) , G = al(Z)E g1and a:(Z U a ) does not exist. Then cp(F) v q(G)= a;(f{Z} U f ( a ) ) and, by computing the rank of both sides in P*, we deduce that f ( a ) E u;(f({Z}). If f ( a ) ef{Z}, then there exists D E 9’; such that f ( a ) E D cf{Z}U f ( a ) ; from (b), D =f{S} with S E 9’: and a E S c Z U a, contradicting the fact that a:(Z U a ) does not exist. Therefore f ( a ) ~ f { l hence }; q ( f { Z } ) dominatesf(F), f(G) and thus F, G, i.e. F v G exists. 0

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Corollary 7.16. With the notations of Theorem 7.10, and, following Corollary 7.11, let m be the integer such that %, 3 ' , have the m-union property. Then, the bouquet C ( & , g2,f) has the m'-union property for some m ' , 1 6 m' s m.

Proof. It follows from Claim 7.8(iv) applied to the strong map Q, defined in Theorem 7.15 after noticing that C('3:, 3 ' ;, f *) is a bouquet of m matroids. 0 Let us describe in more detail the mapping cylinder operation. Let P = C(P,, P2, f ) be the mapping cylinder obtained from PI, P2, f as in Theorem 7.13. Suppose that P2 is a bouquet of m geometric lattices of rank r, with maximal elements zl, . . . ,,z and with least element 02. From the definition of the order relation < c , P is also a bouquet of m geometric lattices with maximal elements z,, . . . , 2,; its rank is r + 1, its atoms are Oz (which is in fact a universal atom of P) together with the atoms of PI and its least element is the least element O1 of PI. Furthermore, if one deletes the upper interval [O,, 4)from P, one obtains exactly the poset P l , i.e. PI = P - [O,, 4) can be realized as the bouquet P with one interval deleted. As a consequence, we have results 7.19, 7.20, 7.21. Next, we give some precisions on how to define the mapping cylinder as a bouquet of matroids, i.e. in set theoretical terminology.

Remark 7.17. Let 9 be an IS on X and '32E 2(9)such that % < '32. From Theorems 7.2, 7.13, the poset P = C('3*,9&, u2) is a bouquet of geometric lattices. Let w be an arbitrary element that does not belong to X. Then one can define P as a bouquet of matroids on X U w whose flats are exactly the sets G E '3, or G U w for G E $I2. Hence, assuming that the O-flat of g2is 0, the set F, = { w } is a universal l-flat of C('31,Yl2, 02);we keep these notations in the remaining of the section. Notice that this amounts to the embedding of the bouquet g1of rank r on X in the bouquet C(%l,9&., u,) of rank r + 1 on X U w. Hence, the mapping cylinder operation is closely related to the notion of embedding of geometries and, also, as noted in [23], to the notion of single element extensions of matroids. We give for illustration an example. Example 7.18. Let %,, %$ be the bouquets of matroids on [I, 61 whose flat configurations are shown below. They have the same IS and % < '3z holds. We picture below the bouquet of matroids C( (92, u2).

Bouquet Yi1:

Bouquets of geometric lattices

Bouquet

305

%2:

1

Bouquet C(% g2,u2):

Proposition 7.19. Let 9 be an IS on X and g1< g2E 2($)be two bouquets of, respectively, m l , m2 matroids; hence m2 ml. Then C(%, s2,u2)is a bouquet of m2 matroids and = C(%, 92,u2) - [F,, 4). Corollary 7.20. Let 9 be an I S having the m-union property and assume that 2(9)is a lattice with greatest element %*. Then, for all % E 2’($), C(%, %*, a*)is a bouquet of m matroids and %= C(%,%*, a*)- (F,, 4); i.e. any bouquet with the m-union property can be realized as a bouquet of m matroids with one upper interval deleted. We deduce in particular from Corollary 7.20 that any bouquet with the 2-union property can be realized as a bouquet of 2 matroids with one less upper interval. We also deduce that any geometric semilattice can be realized as a matroid with one interval deleted, thus restating the “realization” part of Theorem 3.2 [23]. Remark 7.21. We obtain an alternative proof for Theorem 3.5 in the design case: if % is a transversal matroid design with PMD support A, then, since the projection p 1 is a surjective rank preserving strong map (Theorem 7.6), C(%, A, p l ) is a matroid and, from Proposition 6.11, %= C(%,A, p l ) - [F,, -0) is therefore a geometric semilattice.

8. On the shellability of bouquets of matroids

To any poset P, one can associate a simplicia1 complex A(P), called its order complex, whose simplices are the maximal chains x1 <x2 < - - * <x, of elements of P. Recall that a simplicial complex is exactly an IS in which all singletons are independent sets, the simplices correspond then to the independent sets of the IS; the notation of simplicial complex being more specifically used in topological or

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geometrical context. For any simplex m of a simplicial complex A , one denotes by m the subcomplex of A formed by all subsets of m ;its dimension is one less than its cardinality. The dimension of A is the maximum dimension of its simplices, A is called pure when all maximum simplices have the same dimension. Similarly, an IS is pure when all its bases have the same cardinality which is then called the rank of the IS. Let A be a pure ddimensional simplicial complex with vertex set X, = n. A shelling of A is a special ordering of the maximal simplices of A which is favourable for induction arguments. Then, A is said to be shellable if it admits a shelling order. The ordering: m m2, . . . ,m,of the maximum simplices of A is a shelling order if:

1x1

for all i, j , 1=z i <j G s, there exists k, 1d k <j , and x E m,such that: rn, rlm,E m, rlmk= m,- { x }

(8.1)

This amounts to saying that the subcomplex m, rl U: m,is a pure complex of dimension d - 1. The distance between two maximum simplices m,m' is the length k of a shortest simplicial path m = mo, ml, . . . ,mk = m' where the m,'s are maximum simplices such that m,rl mt-, is a (d - 1)-simplex for i = 1, . . . , k; if no such path exists, then the distance between m,m' is a.The diameter of A, diam A , is the maximum distance between any two maximal simplices of A . One says that A satisfies the Hirsch conjecture if diam A d n - d + 1 holds. It is well known that shellable complexes share many combinatorial and topological properties. For instance, an r-dimensional shellable complex has the homotopy type of a wedge of r-spheres (Theorem 1.3, [4]), its reduced homology is known: it vanishes in all dimensions other than r (Proposition 3.10, [2]) and some naturally associated commutative ring is Cohen-Macaulay (for more details, see [3, 41 and references mentioned there). To any bouquet of geometric lattices P are naturally associated two simplicial complexes: its order complex A ( P ) and the complex $ ( P ) of its independent sets of atoms. Similarly, for a bouquet of matroids 3, one considers respectively the complex of chains of flats of %, also called its flat complex and denoted by FL( $3); and its independence system f , also called independence complex. When FL(% is shellable, we also say that 3 is shellable. Note that, as was done by Bjorner for matroids ([2]), one may associate other complexes to a bouquet such as its broken circuit complex; this will be the object of further study in [MI.It is known that when P is a geometric lattice, then both A ( P ) and f ( P ) are shellable ([2, 201); this result was extended to geometric semilattices in [23]. Therefore, it follows from Theorem 3.5 that all well cut transversal geometries are shellable. We now study the shellability of general bouquets. The results presented here come from [18] which will also contain other results of topological nature. Let us mention an application of shellable IS to the study of tight bounds for their reliability polynomials ( [ 5 ] ) . It is obviously not true that any bouquet of matroids is shellable; for a

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counterexample, consider a bouquet whose branches are matroids on disjoint groundsets. In fact, a shellable bouquet must satisfy strong connectivity properties; so, its basis graph must be connected. We will see that, for the case of bouquets of matroids with the 2-union property, this condition is indeed a sufficient condition for shellability. Note that a shellable bouquet must be well cut, which amounts to saying that its IS must be pure.

Proposition 8.2. Let % be a well cut bouquet of m matroids of rank r with roofs X I ,X 1 ,. . . , X,. If its flat complex F L ( q is shellable, then there exists an ordering of the roofs, say X , , X,,. . . ,X,, such that: for all j

B 2,

there exists k, 1s k <j , such that r(X, n Xi) = r - 1 (8.3)

Recall that a maximal chain m of F L ( q is of the form: O S F , ~ . . - S F , with 4 being an i-flat and F, is some roof Xi of %, also called the roof of the chain m ; the length of the chain is: lml= r. The following can be easily verified. Claim 8.4. Let m,m' be two maximal chains of flats of $3 with distinct roofs Xi, Xi. Then, Im r l m'l = r - 1holds if and only if m, m' differ only by their roofs and, then, r(Xi n Xi) = r - 1 holds.

Proof of Proposition 8.2. Consider a shelling order of the maximal chains of F L ( 9 : m l , . . . , m,.We deduce from (8.1): for j 3 2, there exists k, 1s k <j , such that lmk n mil = lmil - 1= r - 1 (8.5) Suppose, for instance, that the first chain ml has roof X , . Let i 3 2 be the first index such that mihas a roof distinct from XI,say mi has roof X,. Then, one deduces from (8.5) and Claim 8.4 that r(Xl fl X,) = r - 1. Let j 3 i + 1 be the first index such that mihas neither roof X I or X,, say mi has roof X,. One deduces again from (8.5) and Claim 8.4 that r(Xl n X,) = r - 1 or r(X, n X,) = r - 1. Clearly, after iteration of this process, one obtains an ordering of the roofs satisfying (8.3). 0

Definition 8.6. Let $3 be a well cut bouquet of m matroids of rank r with roofs X I , .. . ,X,. Its roof graph GR is the graph with vertex set [l, m ] and whose edges are defined as follows: two vertices i, j E [l, rn] are adjacent if and only if r(Xi nXi) = r - 1. Definition 8.7. Let d p be a pure IS of rank r and 53 its family of bases. Its basis graph GBis the graph with vertex set 93 and whose edges are defined as follows: two bases B, B' are adjacent if and only if IB n B'I = r - 1. Two bases B,B' are adjacent in GB if and only if B' is obtained from B by pivoting (or shifting), i.e. by exchanging exactly one element of B by an element

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308

of B'. Hence, the basis graph is connected if and only if any basis can be obtained from any other by a finite sequence of pivots; let us simply recall that pivoting is a fundamental tool in the simplex algorithm for linear programming. For instance, G,, is connected when J? is a matroidal IS. Observe that, for a bouquet ' 3 2(9), ~ its roof graph and its basis graph are closely related; so, for the free bouquet 3 = 3,both graphs coincide and, in general, they are simultaneously connected, as shows Proposition 8.8. Observe also that the diameter of the IS 9 (as simplicia1 complex) coincides with the diameter, diam GB,of its basis graph; therefore, for a pure IS of rank r on X , = n , saying that it satisfies the Hirsch conjecture amounts to saying that diam CB=sn - r holds. It is proven in [20] that the Hirsch conjecture holds for matroidal IS; we extend this result to IS with the 2-union property and with connected basis graph in Proposition 8.16.

1x1

Proposition 8.8. Let 9 be a pure IS. The following assertions are equivalent: (i) the basis graph GBis connected (ii) the roof graph GRof any bouquet % E 6p(J?) is connected.

Proof. The implication (i)+(ii) follows from the fact that, if two bases B, B' contained in distinct roofs X I , X,are adjacent in GB,then i, j are adjacent in GR. Conversely, the implication (ii) 3 (i) follows from the fact that any two bases B, B' contained in roofs X , , X, with r(X, n X,) = r - 1 are connected; for this, take a maximal independent subset I of XI n X,, 111 = r - 1, x E X ,- X,and y EX,A',. Then, from axiom ( I 3 ) , the sets B, = I + x and B, = I + y are bases of J? respectively contained in X,, X, and they are adjacent in G,; now one can connect B to B, in the matroid on XIand, similarly, B' to B, and thus B to B'. tl Proposition 8.9. Let '3 be a well cut bouquet of matroids. If its flat complex FL( '3)

is shellable, then its roof graph is connected or, equivalently, its basis graph is connected.

Proof. Let X I , . . . , X,,, be an ordering of the roofs of '3 satisfying (8.3); one verifies by induction on i 2 2 that i is connected to 1 in GR,henceforth implying that G, is connected. 0 Corollary 8.10. The full d-injection geometry $(N,,. . . , N d ) with lA$l= INdI = n b 2 is not shellable for all d 2 2.

-

=

Proof. Observe that any two distinct bases B, B' of 9(N,, . . . , Nd) are d injective sets of size n satisfying IB f l B'I S n - 2. Therefore the basis graph is totally disconnected. 0

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This fact was observed in [l],Fig. 7.l(b), for the case d = 2. It turns out that, for bouquets of matroids having the 2-union property, the connectivity of the basis graph (or of the roof graph) is enough for ensuring shellability, i.e. the converse of Proposition 8.9 is true. For stating this result, we need another type of poset shellability, introduced in [l], which is favourable for induction proofs. Recall that the length of a poset is the maximum length of the chains of A ( P ) .

Definition 8.11. Let P be a finite ranked poset. A recursive atom ordering of P is defined by induction on the length of P as follows: -if P has length 1, then any atom ordering is a recursive atom ordering -if P has length greater than 1, a recursive atom ordering of P is an ordering al, a2, . . . ,a, of the atoms of P satisfying: for j E [l, t ] ,the poset [aj,4) admits a recursive atom ordering which begins with the atoms that cover some ai for j < i

(8.12)

for j E [2, t ] , there exists i, 1 6 i < j , such that ai v aj exists.

(8.13)

Note that (8.13) is slightly different from axiom (ii) in the original definition of [ l ] ,however both definitions coincide for the case of bouquets of geometric lattices that we consider, also it suffices to adjoin a top element to P for obtaining the original definition of [l] for bounded posets. It is proved in [ l ]that the existence of a recursive atom ordering of P is equivalent to chain lexicographical shellability which implies the shellability of A ( P ) .

Proposition 8.14 (Theorem 7.2, [23]). Let Ce be a geometric semilattice of rank r with IS $ and closure operator a(.). Then, any atom ordering that begins with some atoms 4 = a(xl), . . . ,F, = a@,)such that the set { x l , . . . ,n,} is a basis of 9 is a recursive atom ordering. Proposition 8.15. Let 9 be a pure IS of rank r having the 2-union property. Let 9 be the greatest element of 6p($), so Ce* is a bouquet of 2 matroids with roofs X I ,X,. Let Ce be a bouquet of 9($) with closure operator o(.).Assume that the basis graph G, is connected, then any atom ordering of Ce that begins with some atom 4 = a(xl), . . . , F,-l = a(x,-,) such that the set {xl, . . . ,x , - ~ } is a busis of X I n X 2 is a recursive atom ordering. Proof. We prove the theorem by induction on the rank r of the IS 9 (or of any Ce E 2($)). Let r(.) denote the rank function of 9,then, the rank function of Ce or (8" coincides with r ( . ) on finite rank sets. By assumption, Gsis connected, i.e. from Proposition 8.8, r(Xl nX 2 )= r - 1 and the roof graph GRof Ce is connected. We can suppose that r 3 2.

310

M. Launnt,

M. Deza

We first verify that one can find r - 1 atoms of '3 as in Theorem 8.15. Since r ( X InX,)= r - 1, take a basis 1 = {xl, . . . , qA1} of X1fl X,, then the flats fi = a(xl), . . . , E-, = a ( ~ , -are ~ )atoms of '3 as in Theorem 8.15. Observe that the flat a(Z)= a(&U U &l) is well defined. Define an atom ordering 9 of '3 that begins with the atoms 4 , . . . ,l$-l.We prove that 9 is a recursive atom ordering of '3. We show that Q verifies (8.13). For this, take an atom F = a(x) of '3 which is distinct from Fl, If F = with 2 S i d r - 1, then a(fi U F) is well defined. Else, F is after all E's in the order Q; by applying axiom (14) to the independent sets { x } and 1 (in the matroid on Xi when x E Xi), we deduce that { x , x i } E 9 for some 1 G i s r - 1 and thus u(F U 6)= a ( { x , x i } ) is well defined. We now prove that (8.12) is satisfied. For this, let F be an atom of '3. Then, the intervals [F,4) in '3 and 9 are bouquets isomorphic, respectively, to '3 - F and 9- F, of rank r - 1 and with IS 9 * (X- F) (Theorem 6.5). Furthermore, when F is contained in Xl f l X,, the interval [F,-0) in '3* is a bouquet of 2 matroids X, and the IS 9 (X- F) has the 2-union property; note that its with roofs X1, basis graph is still connected since X1n X, has rank r - 2 in [F, 4).When, for instance, F c X, and F 4 X,, then the interval [F, 4)in %* is a matroid with roof XIand the IS 9 (X- F) is matroidal. Note that the atoms of the interval [F,4) in % are of the form G = a ( { x , y}) with y @ F and {x, y} E 8.Define the set B ( F ) of atoms of [F, 4) in '3 that cover some atom F' of '3 which is before F in the order 9.We show how to construct a recursive atom ordering of the interval [F, 4)in % satisfying (8.12); for this, we distinguish three cases: +

-

e

-

-

Case 1. F c X 1 and F 4 X,. Then, F = a ( x ) with x E X1- X, and, since the independent set I = { x , , . . . , x , - ~ } is contained in X1n X,, from axiom (13), 1 x E 9.The flats GI = a ( { x , xl}), . . . , G,-l = a ( { x , are atoms of [F,-+) such that the set {xl, . . . ,x , . - ~ } is a basis of 8 ; . (X-F). Note that {GI, . . . , G,-l} c B ( F ) holds. Consider an atom ordering of [F, -+) that begins

+

. . . , G,-l and then with the remaining atoms of B ( F ) ; then, with the atoms G1, from Proposition 8.14, it is a recursive atom ordering and it satisfies (8.12).

Cepe2. F G X , ~ X and , F = e for i E [ l , r - l ] . Then the flats G k = a ( { x i , x k } ) for k E [ l , r - 11, k f i , are atoms of [ F , - ) such that the set { x ~ ., . . , xi-1, x i + l , . . . , x , - ~ } is a basis of Xln X, in 8 (X- F). Note that B ( F ) E {Gl, . . . , Gi-l, Gi+l, . . . , G,-l} holds. Consider an atom ordering of IF,+) that begins with atoms of B ( F ) and then continues with the remaining atoms of {Gl, . . . , G i - l ,Gi+l, . . . , Gr-l}; from the induction assumption, this is a recursive atom ordering and it satisfies (8.12). Case 3. F c XIn X2 and F # 6 for i E [l, r - 11. Let F = a@), then, by axiom (14) applied to the independent sets { x } and I = {xl, . . . ,x,-~}, we deduce that

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I - x i + x € 9for some i E [l, r - 11. The sets G k = a ( { x , xk}) for k E [l, r - 11, k # i , are atoms of [ F , 4 ) such that the set I - x i is a basis of X l n X 2 in f (X - F) and {G,, . . . , Gi-,, Gi+,, . . . , Gr-,} E B ( F ) holds. One obtains a recursive atom ordering of [F, 4) satisfying (8.12) by putting first the atoms G,, . . . , Gi-l, Gi+l, . . . , Gr-l, then the remaining atoms of B ( F ) and finally all other atoms.

Proposition 8.16. Let f be a pure IS with the 2-union property and whose basis graph is connected. Then the IS 9 satisfies the Hirsch conjecture. Proof. For two distinct bases B, B’ of f , we denote by d ( B , B‘) the distance between B, B’ in the basis graph; by assumption, it is finite. We first observe that, if B, B’ are bases in a matroidal IS of rank r, then it follows from the basis exchange axiom (B) that: d(B, B’) = r - IB f B‘I. l

(8.17)

Consider now a pure IS f of rank r on X , 1x1 = n, with the 2-union property and connected basis graph. Hence, the greatest element %* of 2 ( f )is a bouquet of two matroids with roofs X,,X, such that r ( X 1r l X,)= r - 1. Let B, B‘ be two distinct bases of $. If B, B’ E Xifor i = 1 or 2, then, from (8.17), d ( B , B’) = r - IB i l B ’ JS n - r. We now suppose that B c X1,B’ E X , and consider elements x E B - X,,x ‘ E B‘ - X 1 . Let I be a maximal independent set such that: B n B’ E I c X1fl X 2 ; then, 111 = r - 1 and, from axiom (13), the sets B , = I + x and B2 = I + x ’ are bases of f respectively contained in X , , X,.From (8.17), we have that: d ( B , B,) = r - IB fl B1l = r - 1- II n B J and d(B’, B,) = r - JB‘n B21 = r - 1- II n B‘I. Using the relations: d ( B , B’) d d(B, B,) + d(B,, B,) + d(B,, B’) and d ( B 1 ,B2) = 1, we deduce that: d (B, B ‘) 6 2r - 1- IZ n B I - ( I n B’l,

(8.18)

For completing the proof, we show that the right hand side of (8.18) is less or equal to n - r. For this, observe that: n 2 IB U B‘ U I1 and IB U B’ U I ( = JBU B‘J+ [ I t - ( ( Bn I ) U (B’ f l I)I = 3r - 1- ( B nI1 - ( B ‘nZI, which therefore implies that: n - r 2r - 1- IB n I1 - IB’ r l II; this concludes the proof. 0

*

The next theorem follows from Propositions 8.8, 8.9, 8.15 and 8.16.

Theorem 8.19. Let f be a pure IS of rank r having the 2-union property, %* be the greatest element of 2 ( f )with roofs X I , X, and % be an arbitrary bouquet of 2(f).The following assertions are equivalent: (i) the basis graph is connected (ii) r(Xl nX,) = r - 1 (iii) the roof graph of % is connected

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(iv) FL( %) is shellable (v) FL(9) is shellable (vi) 9 satisfies the Hirsch conjecture. The shellability of the flat complex F L ( 3 of a bouquet of matroids 97 seems therefore to be an intrinsic property of its IS 9 , i.e. to depend only on properties of $ and not on the flat configuration of the specific bouquet %E 2’($).This is indeed the case for geometric semilattices and bouquets with the 2-union property for which a sufficient and necessary condition for shellability is the connectivity of the basis graph. We conjecure that this is still the case for general bouquets - at least when 9($) is a lattice - so, we conjecture that a bouquet is shellable if and only if the flat complex of its IS is shellable. We address the related open question of finding a necessary and sufficient condition for the shellability of FL($), or Yic

=w).

References 111 A. Bjorner and M. Wachs, On lexicographically shellable posets. Trans. Amer. SOC. 1 (1983) 323-341. 121 A. Bjomer, Homology of matroids, chapter of a forthcoming book on matroids (N. White ed.). [3] A. Bjomer, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. SOC.260 (1980) 159- 183. [4] A. Bjorner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. Math. 52 (1984) 173-212. [S] M.O. Ball and J.S. Provan, Bounds on the reliability polynomial for shellable independence systems, Siam J . Alg. Disc. Meth. 3 (2) (1982) 166-181. [6] P.J. Cameron and M. Deza, On permutation geometries, J . London Math. SOC.20 (3) (1979) 373-386. [7] P.J. Cameron, M. Deza and P. Frankl, Sharp sets of permutations, J. Algebra 111 (1987) 220-247. 181 M. Conforti and M. Laurent, On the geometric structure of independence systems. Math. Prog. Series B (1989) to appear. [9] M. Conforti and M. Laurent, On the facial structure of independence system polyhedra, Math. of O.R., 13 (1988) 543-555. IlO] P. Delsarte. Association schemes and t-designs in regular semilattices, J. Combin Theory A 20 (1976) 230-243. (111 M. Deza and P. Frankl, Injection geometries, J . Combin. Theory B 36 (1984) 31-40. [12] M. Deza and P. Frankl, On squashed designs, Discrete and Computational Geometry 1 (1986) 379-390. 1131 M. Deza and K. Fukuda, On bouquets of matroids and orientation, RIMS, Kokyuroku 587, Kyoto University (1986). [I41 M. Deza and M. Laurent, Bouquets of matroids, d-injection geometries and diagrams, J . Geometry 29 (1987) 12-35. [I51 H. Hanani, On transversal designs, Combinatorics part 1. Math. Centre Tructs 55 (1974) 45-52. f161 M. Laurent, Upper bounds for the cardinality of s-distances codes, Europ. J. Combinatorics 7 f 1986) 27-41. [ 171 M. Laurent, GeomCtries IaminCes: Aspects algorithmiques et algtbriques, University of Paris VII, Doctorat thesis (1986). 1181 M. Laurent, Shellability and related problems for bouquets of matroids, in preparation.

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[19] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes (North-Holland, Amsterdam 1977). [20] J.S. Provan and L.J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. of O.R. 5 (1980) 576-584. [21] D. Stanton, A partially ordered set and q-Krawtchouk polynomials, J. Combin. Theory A 30 (1981) 276-284. [22] D.J.A. Welsh, Matroid Theory (Academic Press, London, New York 1976). [23] M.L.Wachs and J.W. Walker, On geometric semilattices, Order 2 (1986) 367-385. [24] T. Zaslavsky, Extremal arrangements of hyperplanes, Annals of the New York Academy of Sciences, Discrete Geometry and Convexity (1985) 69-87.

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Discrete Mathematics75 (1989) 315-317 North-Holland

315

A SHORT PROOF OF A THEOREM OF V U O S ON MATROID REPRESENTATIONS Imre LEADER Dept. of Pure Mathematics and Mathemutical Statktics, University of Cambridge, 16 Mill Lane, Cnmbridge CB2 lSB, U.K.

In [2] V h o s proved that if a (finite or infinite) matroid is representable over fields of arbitrarily large characteristic then it is representable over a field of characteristic zero. The method of proof is algebraic (see Fenton [l]for further developments of this method). Although the proof of this important result is by no means long, it may be of interest to see how ultraproducts can be used to give a very short and direct proof of it. This is our main aim in this note. We also give a brief proof of another result due to VAmos, namely that if each finite subset of a matroid is representable over an extension field of a given field then so is the matroid itself. Our notation and terminology are fairly standard. A matroid is a pair (M, E ) , where E is a non-empty set system on M satisfying (i) i f A E E a n d B c A thenBEE, (ii) if finite A, B E E with IAI < IBI then A U { b } E E for some b E B\A, (iii) if A E E for all finite A c B then B E E. The elements of E are called the independent sets of the matroid. Thus, in the infinite case, a matroid is just an independence space in the sense of Welsh [3]. A representation of a matroid M over a field F is a map 9 :M --f V, where V is a vector space over F, such that A c M is independent iff the family d ( a ) , a E A is linearly independent in V. Thus in particular 9 is injective on independent sets. Given sets Xi, i E I and an ultrafilter % on I , the ultraproduct of the Xi over %, written Xi/%, is the set of equivalence classes of Xi under the equivalence relation (xi) (yi) if { i E I : x i = yi} E %. We write [(xi)] for the equivalence class of (xi). If the Xi are algebraic structures then n X i / % inherits algebraic operations from the coordinatewise operations on Xi. We are now ready to prove V h o s ' theorem.

n

n

-

n

Theorem 1. Let M be a matroid representable over fields of arbitrarily large characteristic. Then M h representable over a field of characterhtic zero. Proof. For each i = 1,2, . . . , let &: M - t V; be a representation of M over a field I$ with char 4 > i. Let % be a non-principal ultrafilter on N, and set F = He/% and V = %/%. Then it is easy to see that F is a field, with char F = 0, and V is a vector space' over F. 0012-365X/89/$3.50

0 1989, Elsevier Science Publishers B.V. (North-Holland)

316

1. Leader

Define @ :M + V by @ ( a )= [(91(a))].We claim that @ is a representation of M over F. Indeed, let {#I,. . . ,a ( ” ) }be a finite independent set in M , and suppose that Ao’&(a(”) =O. Write Lo)= [(A!’))], j = 1, . . . , n. Then { i E I: C Av)q$(aG)) = 0) E %, and so {i E I : A:’) = 0 for all j } E %. It follows that A’) = 0 for all j, as required. Conversely, let {d’),. . . , a @ ) }be a finite dependent set in M. Then, for each i E N, there are A:’), . . . , A?) E I;I such that C A:”&(ao’) = 0, with A?) # 0 for some j . Put A(‘’ = [(A?)], I = 1, . . . , n. Then C Ao)+(ao)) = 0. Moreover, if Ao) = 0 for all j then {i E I : A!” = 0) E 011 for all j , so that for some i we would have A?) = 0 for all j . Thus Lo)# 0 for some j . 0

c

The proof of the following result, which is also from [2], is just as simple.

Theorem 2. Let M be a matroid and F a field such that each finite subset of M is representable over an extension field of F. Then M is itself representable ouer an extension field of F. Proof. Let I denote the collection of finite subsets of M. For each S E I there is a mapping & : M-, V,, where V, is a vector space over an extension field Fs of F, such that $.5 S is a representation of S over Fs. ForSEI, l e t $ = { T E I : S c T } . Thesets$SEIgenerateafilteronI. Let % be an ultrafilter on I extending this filter, and set F’ = Fs/% and V = V,/%. Then F’ is an extension field of F, and V is a vector space over F’.

I

n

n

Define @ : M + V by $ ( a ) = [ ( $ s ( a ) ) ] .As in the proof of Theorem 1, it is easy to verify that c$ is a representation of M over F’. 0 We remark that it is certainly not the case that if each finite subset of M is representable over F then M is itself representable over F. For example, let M be the matroid on an uncountable set whose independent subsets are those with at most 2 elements. Then any finite sybset of M is representable over Q, but M itself is not. However, the assertion does hold if F is finite. Indeed, if F is finite and each Fs = F then we have F‘ = F i n the proof of Theorem 2. We wish to point out that Theorems 1 and 2 can also be proved by making use of the Compactness Theorem for first-order logic. However, the method of making the theory of the representations of a given matroid into a first-order theory is rather unnatural (it involves fixing an arbitrary basis of the matroid), and we feel that the ultraproduct proofs are simpler and more direct.

References 1 1 1 N.E. Fenton, Matroid representations - an algebraic treatment, Quart. J . Math. (Oxford) 35 (1984) 263-280.

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[2] P. VBmos, A necessary and sufficient condition for a matroid to be linear, in Mobius Algebras-Proceedings of a Conference (Crapo, H. and Roulet, G . , eds.), University of Waterloo (Waterloo, Ontario, 1971) 166-173. [3] D. Welsh, Matroid Theory (Academic Press, London, 1976) xi + 433 pp.

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Discrete Mathematics 75 (1989) 319-325 North-Holland

319

AN ON-LINE GRAPH COLORING ALGORITHM WITH SUBLINEAR PERFORMANCE RATIO Lgszld LOVASZ Department of Computer Science, Eotvos University, Budapest, Hungary and Department of Computer Science, Princeton Universily, Princeton, NJ 08544, U.S.A .

Michael SAKS Department of Mathematics and RUTCOR, Rutgers University. New Brunswick, NJ 08903 and Bell CommunicationsResearch, Morrktown, NJ 07960. Supported in part by NSF grant DMS87-03541 and Air Force Ofice of Scient$c Research grant AFOSR-M71.

W.T. TROTTER Department of Mathematics, Arizona State University, Tempe, Arizona 85287. Research supported in part by NSF grant DMS 87-13994. One of the simplest heuristics for obtaining a proper coloring of a graph is the FErst-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integeF as the color set, assign to each successive vertex the least integer possible (keeping the coloring proper). This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with aU edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and once assigned, the color cannot be changed. The performance function, p,(n), of an on-line algorithm A is the maximum over all graphs G on n vertices of the ratio of the number of colors used by A to color G to the chromatic numbers of G. The Ffrst-Fit algorithm has performance function n/4. We exhibit an algorithm with sublinear performance function.

1. Introduction One of the simplest heuristics for obtaining a proper coloring of a graph is the First-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integers as the color set, assign to each successive vertex the least integer possible, subject to maintaining a proper coloring. This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with all edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and, once assigned, the color cannot be changed. If A is any graph coloring algorithm then, for a graph G, xA(G) denotes the number of colors that A uses to color G. The performance ratio of A on G, denoted pA(G), is xA(G)/x(G),i.e. the ratio of the number of colors used by A to the number of colors in an optimal coloring of G. The perfonnance function of 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

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et

al.

A , denoted pA(n), is defined for each integer n to be the maximum of pA(G) over all n vertex graphs. Various researchers (see e.g. [1,7]) have shown that no on-line algorithm A has bounded performance function on the class of all graphs; indeed, for any on-line algorithm A, there are trees on n vertices for which that algorithm requires at least 1 +logzn colors. On the other hand, p a ( n ) S n for any algorithm. First-Fit does not do much better than this trivial bound even on the class of bipartite graphs since for any integer k, there exists an on-line bipartite graph on 2k vertices for which First-Fit requires k colors (the graph is the bipartite complement of a perfect matching). Recently, Szegedy [19] showed that for any on-line algorithm A and integer k, there is a graph on at most k(2k - 1) vertices having chromatic number k, but for which the algorithm A requires 2k - 1 colors. Thus the performance function for any on-line algorithm A grows at least as fast as n/(logn)’. It is natural to ask whether there is any on-line algorithm that has a sublinear performance function. In this note we settle this question in the affirmative by proving:

Theorem 1. There exkcits an on-line coloring algorithm Color with pcolor(n)= @/log* n)(l + o(1)). Note that in this paper all logarithms are taken to the base 2, and, as usual, log* n is the smallest k for which the k times iterated logarithm, logtk)n = log . log n is at most 1. The algorithm Color is constructed recursively from an algorithm Partition* that partitions the vertex set into subsets each having clique number strictly smaller than the input graph. This kind of recursive construction was used by Wigderson ([21]) to obtain a polynomial (but not on-line) approximate coloring algorithm with performance ratio n(log log n)2/(log n)*. Previous researchers have considered the behavior of on-line coloring algorithms on restricted classes of graphs. The performance function of an algorithm A with respect to a class G of graphs p A ( n ; G ) , is the maximum of pA(G) over all n vertex graphs in the class G.It is an easy exercise to construct an on-line algorithm that achieves a performance function of O(log, n) on the class of bipartite graphs, which is optimal by the lower bound for trees mentioned above. For the class of interval graphs, Kierstead and Trotter [14] showed that there is an on-line algorithm A with performance ratio 3 and this is best possible. Recently, Kierstead [12] showed that First-Fithas bounded performance ratio on interval graphs, solving a question posed by Woodall ([22]) and Chrobak and Slusarek. GyBrfBs and Lehel ([7]) showed that First-Fit achieves bounded performance ratio on split graphs, complements of bipartite graphs and complements of chordal graphs. On-line algorithms have been investigated in the context of several combinatorial optimization problems, including various problems related to dynamic data structures [2,3,8,10,16,18,20], task systems and server problems

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[4,15,5,6], bin packing [9], partitioning a partial order into chains [ll], and representing a partial order by vectors in d-space [13]. We believe that many other problems can and should be analyzed from the perspective of on-line algorithms.

2. An on-line graph coloring algorithm An on-fine graph is an undirected graph G on a totally ordered vertex set V. For any vertex v, let [v] denote the set of vertices preceding or equal to v. The pre-neighborhood of a vertex v, N - ( v ) is its neighborhood in [v].The pre-degree of v, d - ( v ) is the size of its pre-neighborhood. An on-line graph partitioning algorithm is a procedure that constructs a partition n o f the vertex set of G by considering each vertex of G in order and assigning it to one of the previous blocks or creating a new singleton block, without reassigning any of the previously assigned vertices. Such an algorithm is a coloring algorithm if it partitions the vertex set into independent sets. More formally, an on-line partitioning algorithm is a map which associates to each graph G in a class G a partition nGof the vertex set, in such a way that for each graph G E G and each vertex v , the partition of [v] induced by nGdepends only on the graph induced by G on [v]. The on-line coloring algorithm given here is constructed recursively from an on-line algorithm called Partition(n, d ) , where n is an upper bound on the total number of vertices and d is a positive integer bounded above by n . This algorithm partitions the vertex set V of the on-line graph G into sets D1, 4,. . . , Dd, C1,Cz,. . . , C,. Each set 0;is independent and is called a first-fit set, and each set Ciis contained in the neighborhood of some vertex and is called a residual set. Let el = d / n and for i > 1, let ci= &J2. Thus 21-1

.=2($)

.

Say that a subset S of vertices of size s is legal if the intersection of the pre-neighborhoods of its members has size at least esn. Partition(n, d ) is defined as follows. Initially r = 0 and D1, . . . , Dd are each empty. For each arriving vertex v: if v U Diis independent for some i then add v to such 0;. Otherwise if Cj u v is legal for some residual set C j , add v to such a set having maximum size. If there is no such set, increase r by 1 and let C,= {v}. The key property of the algorithm is:

Lemma 2. For n 3 4 and d 5 nllog log n, at most 4n/log log n residual sets are created by Partition(n, d ) on input of any graph having'n or fewer vertices.

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We begin with a simple combinatorial lemma.

Lemma 3. Let 6 be a constant with 0 < 6 < $. Let S,, &, . . . ,S, be subsets of a set S such that IS,(3 S IS1 for all i and IS, n Sj(< S2 (S(/2, for i # j . Then q < 216. Proof. Suppose, to the contrary that q 3 2/6 and let j = 12/61. For 16 i sj, let Then the sets T,, T2, . . . , I;. are disjoint and = S, - (S, U S,U * - U S,-,). > 6(1- S(i - 1)/2) IS[. Hence 17J 3 ISi[ - IS, n S,l - IS, n &I - . - - IS;n IS1 5 I& U GU

- - - U T,l=

+

a contradiction, establishing the lemma.

- - - + 11;.1> Sj(1-

6 ( j - 1)/4) IS1 > 16,31,

0

Lemma 4. Let G be an on-line graph with at most n vertices. Then when , sets have size t. Partition(n, d ) terminates, at most 2 / ~residual Proof. For each j E (1, . . . , r } let Aj be the intersection of the preneighborhoods of all of the vertices in Ci. By the definition of the algorithm, if Cj has size f , A, has size at least E,n. Furthermore, we claim that if two sets C, and Cj both have size t , lAinAjl < E,+,n = n ~ : / 2 ,which by Lemma 3, with 6 = E , finishes the proof. To prove the claim, let v be the last vertex added to either Ci or Cj and suppose it was added to Ci. Before it was added, Cihad size t and Cj had size t - 1. Since v was added to a set having smaller cardinality than C,, C, U IJ is not legal. This implies that the intersection of the pre-neighborhood of u with Ai has fewer than elements. Since Ai is contained in the preneighborhood of v , the claim is established. 0

Proof o f Lemma 2. For any integer k 3 2, the number of residual sets that have size at least k at most n / k . By Lemma 4, the number of residual sets of size less than k is at most 2 ( 1 / ~+ , l / ~+,- - * + l/ck-,) which is bounded above by 1 / E k . Hence the number of residual sets is at most n& + 1/Ek. Taking k = loglogn/2 yields n / k + l / e kC 2n/log log n + 2(log log n)''Ogn =z4n/log log n. 0 Next, Partition is used to construct a second partitioning algorithm called Partition*. For k 3 2, let n,, be the largest integer such that nk/log,log, nk s 2k (e.g. n 2 = 4 and n 3 = 16). The on-line algorithm Partition* takes as input any graph and produces a union of disjoint partitions as follows: Place incoming vertices in a single class until the first vertex is received that has a neighbor in that class. Starting with that vertex, apply Partition(n2, 4) to the first n2 vertices. Apply Partition(n3, 8) to the next n 3 - n , vertices, and in general, apply Partition(nk,2 9 to vertices { n k - , + 1, . . . ,n k } .

Lemma 5. Suppose Partition* is applied to an on-line graph G on n vertices. If G rC. an independent set then Partition* produces a single class. Otherwise it produces

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at most 20nllog log n sets, and each has chromatic number strictly less than that of G. Proof. The behavior of Partition* on an independent set is apparent from the definition. On a general graph, every set produced is either a first-fit set (which is independent) or a residual set, which lies in the neighborhood of some vertex of G, and thus has chromatic number strictly less than that of G. It remains to bound the number of sets created. Let k be the least index such that 2"-' < n/log log n Q 2". For any i, at most ni - ni-l vertices are partitioned by Partition(ni,2). This results in at most 2' first-fit sets and, by Lemma 2, at most 4ni/loglogni residual sets, for a total of at most 5(2') sets. Thus, the total number of sets created by Partition* is at most (1 5(22) 5(23) 5(24) 5(2")) s 10(2k)Q 2On/log log n sets. 0

+

+

+

+- - - +

Finally the algorithm Color is defined recursively from Partition*: Run Partition* on G. For each class besides the first (independent) class produced, color it by a recursive call to Color. It is easily shown by induction on the number of vertices of G, that Color partitions any input graph into independent sets. Define c(n, k ) to be the maximum number of colors used by Color to color an input graph on n fewer vertices and chromatic number at most k. Obviously c(n, 1) = 1 and c(n, k ) G n. Define h(')(n) = n, h ( n )= h(')(n)= max(1, log log n/20}, and for k b 2, h(k)(n)= h(k-')(h(n)).Note that for all k and n positive, h(')(n) is a concave function of n.

Theorem 6. c(n, k ) 6 n/h("-')(n)for k b 1 and n

1.

Proof. We prove the result by induction on k; the case k = 1 is trivial. Suppose Color is applied to a graph on n vertices having chromatic number k. By Lemma 5, the main call Partition* produces at most n / h ( n ) classes each having chromatic number at most k - 1. Each of these is recursively colored using Color. Thus the number of color classes created can be bounded above by: t

c(n, k ) s

c(ni, k

max tsn/h(n)

- 1)

i=i

by the induction hypothesis. Since n/k@-*)(n)is a convex function of n, the right hand side is bounded above by taking all of the nits to be equal and t to be as large as possible, yielding an upper bound of n/h(n){h(n)/h(k-2)(h(n))} = n/h@-')(n). 0

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Corellrvy 7 . The algorithm Color has a worst case performance ratio of at most (2n/log* n)(l + o(1)).

Proof. By Theorem 6, the performance ratio of color on a graph on n vertices with chromatic number k is most n/kh(k-')(n),which is maximized for fixed n when k is the least index with hCk-l)(n) = 1 (or, possibly 1 less than that) and this index is asymptotically equal to log* n / 2 . 0 Note that the performance ratio of O(n/log(2k-2)(n)) for graphs of chromatic number bounded by k can be improved(!) to O(n logCzke3) ( n ) / l ~ g ( * ~(n)) - ~ ) for k 3 3 by modifying Partition' as follows. Initially, instead of constructing a single independent set, apply a bipartite graph coloring algorithm using at most O(log IVl) colors, switching to Partition(n2,4) only when a vertex is received that cannot be processed by that algorithm. Of course this does not improve the worst case performance ratio over general graphs. Thus for graphs of chromatic number at most 3, there is an algorithm that achieves performance ratio O(n/log log n). On the other hand, the only known lower bound is the O(log n) lower bound for trees mentioned in the introduction (note that Szegedy's lower bound is not useful for graphs of bounded chromatic number). It would be interesting to close the gap.

References [l] D. Bean, Effective coloration, J. Symbolic Logic 41 (1976) 469-480. [2] J.L. Bentley and C.C. McGeoch, Worst-case analyses of self-organizing sequential search heuristics, Communications of ACM, to appear. [3] J.R. Bitner, Heuristics that dynamically organize data structures, SIAM J. Comp. 8 (1979) 82-110. [4) A. Borodin, N. Linial and M. Saks, An online algorithm for metrical task systems, Proc. 19th Annual ACM Symp. on Theory of Computing (1987) 373-382. [S] R.R.K. Chung, R.L. Graham and M. Saks, Dynamic search in graphs, in Discrete Algorithms and Complexity (Academic Press, 1987). 161 F.R.K. Chung, R.L. Graham and M. Saks, A dynamic location problem for graphs, Preprint. [7] (a) GyM& and J. Lehel, On-line and first-fit colorings of graphs, J . Graph Theory, to appear. [8] C. Gonnet, J.I. Munro and H. Suwanda, Toward self-organizing search heuristics, Proc. 20th IEEE Symp. Foundations of a m p . Sci. (1979) 169-174. [9] D.S. Johnson, A. Demers, J.D. UUman, M.R. Garey and R.L. Graham, Worst case performance bounds for simple one-dimensional bin packing algorithms, SIAM J. Computing 3 (1974) 299-325. [lo] A.R. Karlin, M.S. Manasse, L. Rudolph and D.D. Sleator, Competitive snoopy catching, Proc. 27th IEEE Symp. Foundations of Comp. Sci. (1986) 244-254. [11] H.A. Kientead, An effective version of Dilworth's theorem, Trans. Amer. Math. SOC.268 (1981) 63-77. [12] H.A. Kierstead, The linearity of first-fit colorings of interval graphs, SIAM J. on Discrete Math., to appear. (13) H.A. Kierstead, G.F. McNuIty and W.T.Trotter, A theory of recursive dimension for ordered sets, Order 1, 67-82.

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[14] H.A. Kierstead and W.T. Trotter, An extremal problem in recursive combinatorics, Congressus Numerantium 33 (1981) 143-153. [15] M. Manasse, L. McGeoch and D. Sleator, Competitive algorithms for on-line problems, Proc. 20th Annual Symp. on Theory of Computing (1988). [16] R. Rivest, On self-organizing sequential search heuristics, CACM 19 (1976) 63-67. [17] J.H. Schmerl, Recursion theoretic aspects of graphs and orders, in Graphs and Order, I. Rival, ed. (D. Reidel, 1984). 467-484. [18] D. Sleator and R. Tarjan, Amortized efficiency of list update and paging rules, CACM 23 (1985) 202-208. [19] M. Szegedy, personal communication. [20] R.E. Tarjan, Amortized computational complexity, SIAM J. Alg. Disc. Methods 6 (1985) 306-318. [21] A. Wigderson, Improving the performance guarantee for approximate graph coloring, J. ACM 30 (1983) 729-735. [22] D.R. Woodall, Problem no. 4, Combinatorics (Proc. British Combinatorial Conference), London Math. Soc. Lecture Note Series 13, T.P. McDonough and V.C. Marvon, Eds. (Cambridge University Press, 1974) 202.

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Discrete Mathematics75 (1989) 327-334 North-Holland

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THE PARTITE CONSTRUCTION AND RAMSEY SET SYSTEMS Jaroslav NESETkIL Department of Applied Mathematics, Charles University, Prague, Czechoslovakia

and Vojt6ch RODL Department of Mathemah, Czech Technical University, Prague, Czechoslovakia

This paper deals with Ramsey properties of finite set systems of a given type. We present new and simple proofs of some of the most general results in Ramsey theory for set systems. The proofs rely on a new proof of the Partite Lemma which is combined with an amalgamation technique known as Partite Construction.

Introduction The following result [18] is one of the most famous and fundamental of combinatorial statements. Finite Ramsey Theorem. For every choice of positive integers t, a, b there exists a positive integer c such that c -+ (b);.

Here c+ (b); is a short hand notation (due to Erdos and Rado) for the following statement: For every partition of the collection of all a-element subsets of a set X of size c, there exists a b-element subset B of C such that all a-element subsets of B belong to one class of the partition. This theorem has been generalized many times and several of these generalizations are both profound and difficult to prove. Motivated by general results due to Rado [19] and Graham, Leeb and Rothschild [3], one of the main streams of the research was formed by efforts to prove a very general result which would imply all the known (usually difficult) instances. Thus development culminated with the proof of the Ramsey theorem for systems, which we shall state after introducing a few standard notions. A type A = (&; 6 E A) is an indexed collection of positive integers. Throughout this paper we will fix an index set A and a type A. A system A of type A is a pair (X, A) where X is a finite linearly ordered set, A = (&; u E A), and E (As customary, here ( f )denotes the set of all for 6 f 6 ’ . k-element subsets of X.) We shall suppose that & n.Ua,=8 Elements of the sets & are called edges of A.

(c).

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

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A is a subsystem of B = (Y,K) if X is a subset of Y with the induced order, and A& = J\rS r l P(X)for every 6 E A. Two systems (X, A) and (Y, N)are isomorphic if there is a monotone bijection f .X+ Y taking A onto K. A subsystem of B isomorphic to A is called a copy of A in B. Denote by (A") the set of all copies of A in B. A system F is called irreducible if every pair of points of F is contained in an edge of F. The arrow C-+(B):' is defined by analogy with the classical Erdiis-Rado case: C+(B)f if for every partition ( ~ ) = d l U . - - U dthere t exists B ' E ( ~ ) and an i , exists such that ( y ) ) c d i .The following result was proved by the authors in [6] and [7].

The Ramsey Theorem for Systems. Let t be a positive integer and let A , B be systems. There exists a system C such that

c-, (B)f. Moreover, if A , B do not contain an irreducible system F then C may be chosen with the same property. The original proofs of this result were difficult and complex, see [7, 121. Let us remark that related resuits were obtained by Abramson and Harrington and by Prommel [l, 171. However, for some special cases (such as partitions of edges of graphs and hypergraphs) several simple proofs were found, see "13, 141. These proofs are variations on a common theme - the systematic use of amalgamation of partite systems. This Amalgamation technique has been known to authors since 1976 and was effectively used in several papers [lo, 13,14,15]. This technique did not imply the Ramsey theorem for systems. A breakthrough was achieved in 1980 [16], with a proof of an old conjecture of Erdos related to the Ramsey property of rectangle free graphs. (This seemingly esoteric question is a cornerstone of the area and yields e.g. the existence of complex designs; see the discussion below in Section 3). The purpose of this note is to further extend the methods of [14, 161 and to present a new proof of the Ramsey theorem for systems. (H.J. Prommel and B. Voigt recently found a different simple proof of this result. Their method is also a variant of the Amalgamation technique.) It appears that our approach is strong and flexible enough to yield virtually all the known Ramsey theorems for special classes of set systems. The paper is divided in three parts: In the first part we derive the Partite Lemma which is the starting point of our amalgamation technique. This part uses one new trick. In the second part we apply the Partite Construction. This part is routine for anyone familiar with the Partite Construction and we closely follow the ideas of [14]. For the convenience of the reader we outline the proof of the basic properties of the Partite Construction. In the final part we state several strengthenings of the above results which follow from our method.

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1. The Partite lemma An a-partitie system a is a pair ((Xi):=l, A) where (i) X = UfXlXi is an ordered set satisfying X, <X,< - < X, (ii) A = (Aa;6 E A), Aa E (n”,) (iii) J M n X i J e l f o r e v e r y M E A a i, = l , . .. , a , ~ E A .

-

Sets Xi are called parts of A, elements of M edges of A. Property (iii) implies that edges are transversals with respect to the family X,< - * - < X,. Given a subset Y G X we denote by tr(Y) the trace of Y , i.e. the set ( i ; Y n Xi # 0 ) . A is called tramversal if /Xi/ = 1 for every i = 1, . . . , a. A is a subsystem of B= X ) if there exists a monotone injection i : ( 1 , . . . ,a } + (1, . . . . , b } such that XiE for i = 1, . . . , a and

((x)f=l, xli,

.cl,= Xan

(3

for (IE A.

As before ((Xi)tl, A) is isomorphic to ((K):=,, X ) if there is an order preserving bijection from X = Xi to Y = K sending each part X,onto the corresponding and taking A to X . A subsystem of B isomorphic to A is called a copy of A in B; the set of such copies is denoted again by (A”). The arrow notation carries over in the obvious way.

ui

x

ui

The Partite lemma. Let t be a positive integer and let A and B be a-partite systems. Moreover, let A be transversal. Then there exists an a-partite system C such that

C-, (B):. Proof. Put A = ((Xi);=,, A), B = ((K):=l, X). As A is transversal we may suppose without loss of generality that UaEd A, is the set of all subsets of X (this may be achieved by adding a set of “dummy” edges to A and X). Without loss of generality we may also suppose that every vertex y E Y is contained in a copy of A. This is a general comment (see e.g. [9]): if B* is the subsystem of B induced by (2) and C*+(B*):’ then we can easily construct a system C such that C-, (B): by enlarging every B* E (2:) to a system B. Now take N to be a sufficiently large (indeed very large) number; the actual value of N will be estimate later. Define an a-partite system C = ((Zi)bl, O), 6 = (aa;6 E A) as follows. Set Zi= Y;: X - - X % (N times); i.e. each element of Zihas the form (xi; xi E &, j = 1, . . . ,N).Put Z = Uf=,Ziand for j = 1 , . . . , N define a projection arj:Z+ Y by arj(xk;k = 1, . . . , N)= x i . Clearly nj maps 2, to Yk. We define B = (Oa; 6 E A) as follows. First put Xa= ./\rb U N,where XL is the set of all edges of N which belongs to a copy of A in B, N,= X’ - X; (note that we cannot assume that = 0).

-

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330

The set {(xf; j = 1, . . . , N); k = 1, . . . , na} belongs to 0, if tr({xf; k = 1, . . . , R ~ } )= tr({xf; k = 1, . . . , na}) for all j , j ‘ s N and one of the following possibilities occurs: (1) {xi”; k = 1, . . . , na} E X i for every j = 1, . . . , N ; (2) there exists a non-empty set o G (1, . . . , N} and a set { x k ; k = 1, . . . , R ~ E} NL such that {xf; k = 1, . . . , R ~ = } { x k ; k = 1, . . . , na} for all j E o and for all j @ o there is some q with

{ ~ fk;= 1, . . . , Q } E N ; . (Note that, in general, q # 6, however q is determined by tr({xf; k = 1, . . . , ns}). We shall prove that C - , ( B ) f provided N is large enough. This follows easily from the following two facts. Fact 1. A‘ E (2) iff nj(A’)E

(A”)

for every j = 1, . . . , N .

Proof. Obvious from the definition of 0. 0 In order to state Fact 2, we introduce some notation. Put (2) = { A l , . . . ,A r } . Put R = (1, . . . ,r}. Think of the set R N endowed with Hales-Jewett (combinatorial) lines. A line is a set L of the following form: Fix w c (1,. . . , N} and a’= (at,. . . , & ) e R N and put ~ = { ( ,a. ,. . , aN); ai=a4 f o r i e o ,

aj=aj fori,jEo.

Clearly ILJ= r. Given a = (a1, . . . , aN)E R N denote by V ( a ) the set of all V(a). By virtue of vertices x of Z which satisfy nj(x)E A , . Put V ( L )= UaEL Fact 1 the set (2) is in 1-1 correspondence with the set R N . Fact 2. Let L be a line of RN. Then V ( L ) induces a copy of B in C.

Proof. Check the definition of V ( L ) . 0 Now we invoke the classical Hales-Jewett theorem [5] and choose N sufficiently large such that for every partition of R N into t classes one of the classes contains a monochromatic line. This implies C + (B):. Indeed, let (2) = .dlU * U d,be a partition. By Fact 1 this induces a partition R N = .d; U U Sg: by (Y E d,! iff V ( a )induces a copy belonging to di.By the Hales-Jewett theorem there exists a monochromatic line I in RN. This in turn, using Fact 2, B ’ E (g) such that ( 5 )is monochromatic. 0

-

--

2. The Partite construction

In this part we prove the Ramsey theorem for systems by means of the Partite Construction. We follow closely the construction given in [14, 131.

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Let k, A, B be fixed. Consider A as a transversal a-partite system and B as a transversal b-partite system. Explicitly

B = ({Yl, * * * ,Y b } , 4. Put p = r(a, t, b) = min{n; n+ (b):} (the classical Ramsey number). Put q = (z), ( { ' y p ) ) = {MI, . . . ,M q } . We shall construct "pictures" Po, . . . ,P k , . . . , Pq by induction on k. Picture Pq will be the desired system C. Let P o = ((Xp)ip,l, 0") be a p-partite system where for each choice of b parts . . . , the subsystem of Po induced by them contains a copy of B. Such a "picture" P" may be formed as a disjoint union of copies of B. Suppose pictures P k = ((Xf)f=',Ok), k < q, be given. Consider Mk+l and the a-partite system Dk+l induced in P k by parts Xf where yi belongs to Mk+'. By the Partite lemma there exists an a-partite system Ek+' such that

x,

x,

Ek+'_* (D"")f. Extend each copy of Dk+' in Ek+' to a copy of Pk in such a way that the distinct copies of Pk intersect in vertices of Ek+l only. In this amalgamation the parts of distinct copies of P k are preserved. Denote the resulting amalgamation by ($:)*Pk. (If a more explicit definition is needed, see [15]). Put Pk+'= ($;) * Pk. Finally, put C = Pq. We claim that C has the desired properties.

Claim 1. Every irreducible subsystem in C is a subsystem of B. Proof. Induction on k. This being trivial for k = 0, in the inductive step the amalgamation does not create any new irreducible system. 0 Claim 2. C+ ( B ) f . Proof. We apply backward induction on k = q, q - 1, . . . , 1. In the inductive step (k 1+ k) we apply the Partite lemma and find a copy of Pk such that all copies of A with trace Ak are monochromatic. this leaves us, for k = 0, with a copy P' of Po in C where the color of a copy of A in P depends only on its trace. However, such a copy of Po contains a monochromatic copy of B by the construction of P" and the fact that p _* (b):. 0

+

3. Applications The Partite construction yields, in the spirit of [9] and [ll], several results stronger than the Ramsey theorem for systems. We list some of them. A. Horn-connectedgraphs

First we give an auxiliary definition. Let B = (X, 4)be a set system. A set Y c X is called a cut of B if there is a partition of X - Y into two disjoint sets

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332

Y,, Y, such that no pair {yl, y2}, yl E K, y2 E Y2,is covered by an edge of B. We shall also consider the cut Y as a subsystem of B determined by Y. A system B is called homA-connected if no cut Y of B has a homomorphism into A. Here a homomorphism is an edge-preserving mapping. (The notion of a Horn Kkconnected graph coincides with the notion of chromatically k-connected graph, see [15].) It is also convenient to recall the following notion from [9]: Given a (possibly infinite) set 9 of systems, denote by Forb(9) the set of all those system A which do not contain any system F E 9 as a weak subsystem of B. (A is a weak subsystem of B if every vertex (edge, respectively) of A is a vertex (edge, respectively). Now we have the following result. Theorem 3.1. Let 9 be a set of homA-connected systems. Then for every positive t and every B E Forb(.@ there exim C E Forb( 9) such that C - (B):.

Proof. First fur P = ( T , 9)such that P - + ( B ) f . Put T = (1,. . . ,p}, (;)= { B , , . . . , B 4 } . Define picture P = ((X)iP,,, A') by:

x;= {i} x (1, . . . , q } and iff {(i,, m j ) ; j = 1, . . . , n b } E k = mj = mjvforj, j ' = 1, . . . , q, {ij;j= 1, . . . , nb} is an edge of B and belongs to Nb. (Thus Po is a disjoint union of copies B1, . . . , B, with traces induced by (i)). Clearly Po E Forb(9) and the amalgamation does not create any new homA connected subsystem of Pk+'.(Note that every a-partite system D', . . . , 0 4 may be homomorphically mapped into A.) 0 The Partite Construction is very convenient for constructing sparse Ramsey graphs. This is not surprising as one of the byproducts of the partite construction is a new easy construction of highly chromatic graphs without short cycles [lo]. There are various ways of defining sparseness and we list them in the order of increasing difficulty. 0. Sparse Ramsey theorems - Ramsey families

We say that 3 c (2) is a Ramsey family if for every partition

(;)=dlu..

.Ud,

there exists B' E 3 such that (%)c di for some i. We denote this by C 4 (B):

Partite construction and Ramsey set systems

333

We associate with 8 a uniform hypergraph H i = (X, E) where X = (2) and

E=

c)

[

; B E 8).

We have the following result for sparse Ramsey families.

Theorem 3.2. For A, B and positive integers t, 1 there exist C and a system 48 c (2)such that (1) c 3 (B):' (2) The hypergraph H i has no cycles of length s l . Proof. We proceed by induction on 1 and construct Q by means of the Partite Construction (see Section 2). Put 8 '= In the inductive step we assume that there exists a system Bk+'5 (:"*'.) such that H'&+I has no cycles of length s l - 1. We form the picture Pk+'= a'+'* Pk.Assuming that in Pk we have a system sek then in Pk+'we may define a system Bk+' as gk+'* Qk consisting of copies of B which have no cycles of length SZ (it appears that all copies in $23' and thus sek+' are transversal). See [15] where this argument is covered in details. 0

(z).

C. Sparse Ramsey theorems - cycles in copies We prove the following result concerning sparse copies.

Theorem 3.3. Let Be be a Hom A-connected system. Let t, 1 be positive integers. Then there exists a set system C with the following properties: (1) C+(B);4 (2) The hypergraph H a ) has no cycles of length Sl. Proof. We apply the Partite Construction. We start with picture Po introduced in the above proof of Theorem 3.1 We proceed by induction on k. In the inductive step we use Theorem 3.2 to obtain a Ramsey system Bk+' c ($:) without cycles of length
D . Linearity We say that a system B is A-linear if any two copies of A in B intersect in at most one vertex. Typical examples of linear systems are Steiner systems. In [16] we proved a Ramsey theorem for Steiner systems. More generally, we have the following.

Theorem 3.4. Let A be a system t positive integer. Then for every A-linear B there exists A-linear C such that

C+ (B):'.

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J. Ndetfil, V. R d I

Check the construction in the Proof of the Partite lemma. Use the fact that the Hales-Jewett lines form a linear system. This implies that two copies of B in C intersect either in a copy of A or in a subset of one part of C. Use this fact in Partite Construction: as all copies of B in all pictures are transversal, prove linearity of B by induction on Po, P', . . . , Pk. 0

proof.

References [l] F.G. Abramson and L.A. Hamngton, Models without indiscernibles, J. Symbolic Logic 43

(1978) 572-600. [2] P. ErdBs and A. Hajnal, On chromatic number of graphs and set systems, Acta Math. Acad. Sci. Hung. 17 (1966) 61-99. [3] R. Graham, K. Leeb and B. Rotschild, Ramsey theorem for a class of categories, Advances in Math. 8, 3 (1972) 417-433. [4] L. Lovhz, On chromatic number of finite set-systems, Acta Math. Acad. Sci. Hung. 19 (1968) 59-67. [5] A.W. Hales and R.I. Jewett, Regularity and positional games, Trans. Amer. Math. SOC.106 (1%3) 22-229. [6] J. Nektfil and V. Riidl, A structural generalization of the Ramsey theorem, Bull. Amer. Math, SOC.83, l(1977) 127-128. [7] J. Nektfil and V. Rodl, Partitions of relational and set systems, J. Combinat. Theory B (1977) 289-3 12. [8] J. NeSetfil and V. Rodl, Partitions of vertices, Comment. Math. Univ. Carol. 17, 1 (1976) 85-95. 191 J. NeSetfil and V. Rodl, Partition (Ramsey) theory- a survey, In: Coll. Math. SOC.J6nos Bolyai, 18. Combinatorics (North-Holland, 1978) 759-792. [lo] J . NeZetfil and V. Rodl, A short proof of the existence of highly chromatic graphs without short cycles, J. Combinat. Theory B 27 (1979) 225-227. Ill] J. Nektfil and V. Rodl, Partition theory and its applications, In: Surveys in Combinatorics (Cambridge Univ. Press, Cambridge, 1979) 96-156. [12] J. NeSetfil and V. R d l , Ramsey classes of set systems, J. Combinat. Theory A 34, 2 (1983) 183-201. [I31 J. Nektfil and V. Riidl, Two proofs of the Partition property of set systems, Europ. J. Combinat. 3 (1982) 347-352. (141 J. Nektfil and V. Rodl, A short proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica 1 , 2 (1981) 199-202. 1151 J. NeSetfil and V. Riidl, Sparse Ramsey graphs, Combinatorica 4, 1 (1984) 71-78. [16] J. Nektfil and V. Rodl, Strong Ramsey theorems for Steiner Systems, Trans. Amer. SOC.303, 1 (1987) 183-192. [17] H.J. Prommel, Induced partition properties of combinatorial cubes, J. Combinat. Theory A 39 (1985) 177-208. [IS] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930) 264-286. 1191 R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933) 424-480. [ZO] J. NeSetiil and V. Riidl, Partite construction and Ramsey space systems (to appear).

Discrete Mathematics 75 (1989) 335-342 North-Holland

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SCAFFOLD PERMUTATIONS P. ROSENSTIEHL Centre d 'Analyseet & MathPmatique Soci.de, 54, Boulevard Raspail, 75006 Paris, France

The so-called in-order of the vertices of a binary tree T, as used in the data structure literature, is defined by the rule that each vertex follows all vertices of its left subtree, and precedes all those of its right subtree. Now, label the n vertices of T from 1 to n according to the shelling order, the root being labelled 1, the vertices at the same level being labelled consecutively from left to right, and among two levels the one closer to the root being labelled first. The permutation defined by taking the labels in the in-order of the vertices of T is called a scaffold permutation. The scaffold permutations on [l, n] are shown to form a code for the set of unlabelled binary trees with n vertices. The generalization of the scaffold property to bipermutations on [l, n] generates a code for the set of rooted maximal planar maps with n + 3 vertices. ScaEold permutations and bipermutations afford some special features in the computation of their table of inversion, which is relevant to the Fraysseix-Pollack algorithm for embedding a planar map in the grid.

1. Introduction Given a family of graphs, it is an old challenge to define a code of words whose letters are the vertices of the graphs. The Priifer code is a famous one for labelled trees with n vertices. Here we give such a representation for unlabelled binary trees and rooted maximal planar maps. The problem arises from new efforts to draw maps, and specially from the recent work of Fraysseix, Pach and Pollack [l] on F6ry embeddings in the grid. It is also connected with some elementary data structures, and in particular the splay trees of Sleator and Tarjan [4]. We first recall some elementary definitions. A binary tree T here denotes a finite rooted tree in which each vertex has at most two children, a left child vertex or none, and a right child vertex or none. The left (resp. right) subtree of the vertex v of T is the subtree of T whose root is the left (resp. right) child of v, if it exists. The in-order of T is the total order of the vertices of T such that each vertex follows all those of its left subtree, and precedes all those of its right subtree. The pre-order of T is the total order of the vertices of T such that, for vertex v, those of its left subtree are smaller than those of its right subtree, and all smaller than v. A consktent labelling of the n vertices of T is a labelling from 1 to n consistent with the partial order of T, the root being, as the minimum, always labelled 1. It is a fact that there is a one-to-one mapping between the n! permutations on [l,n] and the consistently labelled binary trees with n vertices. The consistent labelling in which the vertices of the same level are labelled consecutively from left to right (see Fig. 1) will be called the shelling order of T. 0012-365X/89/$3.50 C(J 1989, Elsevier Science Publishers B.V.(North-Holland)

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9

5 4

8

3

2

2

1

5

9 7

4 4

3 2

5

6

2

3 1

1 & =

8

2

4

1

5

9

7

3

6

8

Fig. 1 . A scaffold permutation and its binary tree.

A permutation t on [l, n ] is considered here as a word of n occurrences, with one occurrence of each element of (1, n]. Given a binary tree T with its vertices labelled in the shelling order, the permutation defined by taking the labels in the in-order of the vertices is by definition a scaffold permutation of order n (see Fig. 1). A similar permutation appears in the literature, the so-called Catalan permutation, defined by the in-order of the vertices labelled according to the pre-order of T. A bipermutation 0 on [l, n ] is a word of 2n occurrences, with two occurrences of each element of [l, n ] . A bipermutation on [l, n ] is said to be consistently nested if, whenever an element x has one occurrence between the two occurrences of an other one y, it has both, and x is smaller than y . A bipermutation where the two occurrences of each element are consecutive is said to be simple. A simple bipermutation is equivalent to a permutation; therefore permutations can be considered as a particular case of bipermutations. The table of inversion p of a bipermutation (I on [1, n ] , as defined by Knuth [3], gives for each occurrence of x in [l, n ] , the number p ( x ) of Occurrences greater than x placed in t before x . A binary graph B is a rooted oriented acyclic plane graph in which each vertex has at most two children, a left child vertex or none, and a right child vertex or none, and an indegree of at most two. By rooted is meant that a unique vertex has indegree zero. Each other vertex has one or two fathers, a left father and a right father. Turning round a vertex clockwise we meet the left son, the right son, the right father and the left father, in that order (see Fig. 2). The shelling order of B requires a proper definition of the level of the vertices of B (see Fig. 2). Define

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337

the level 1 as the root alone, which is labelled 1. The levels being defined up to k - 1, the level k is defined as follows. A vertex with indegree 1and whose father is on level k - 1, is put on level k. A vertex v with indegree 2 is put on level k if both its fathers are on levels less than k, and if the vertices that it “encompasses” are on levels less than k. Here the vertices “encompassed” are those belonging to the face of the plane graph B defined by the angle of the two indegree incidences of v taken counterclockwise from the left father to the right father of v. Now on level k, the vertices are ordered from left to right, that is according to a clockwise search from the root. The binary tree T ( B ) is derived from B by splitting any vertex v of indegree two, into a left copy carrying the left indegree and the left

88

6

99

7

5

7

4

66

55

44

3

33

2 11

1

,,

=

22

88

7

11

44

66

33

7

99

55

1

Fig. 2. A scaffold bipermutation and its binary graph.

P. Rosensiiehl

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outdegree, and a right copy, carrying the right indegree and the right outdegree; associate with a binary graph B and its shelling order, the binary tree T ( B ) where the two copies of a split vertex keep the same shelling label. We now construct a permutation associated with T ( B ) by following the rules already given for constructing the scaffold permutation of a binary tree, but keeping the vertices on the levels already assigned. Any element now appearing only once (corresponding to a vertex of indegree less than 2) is doubled. The bipermutation obtained is by definition a scaffold bipermufation of order n (see Fig. 2 ) . A planar map M is said to be maximal if each face of M has a triangular boundary. It is considered rooted by the choice of one edge (a, b) and an orientation from a to b. The face on the right of (a, b), whose other vertex will be called c, is taken as the infinite face, and the one on the left of (a, b) has a third vertex labelled 1. The shelling order of M requires a proper definition of the levels of the vertices of M (see [l] and Fig. 3). Define the level 1 as the vertex 1 alone (a and b can be regarded as being at level 0). The levels being defined up to C

7 is the right son of 2 7 is the left son of 5 = 228871144663379955 Fig. 3. A scaffold bipermutation u and its maximal planar map. (The associate binary graph B ( M ) is on Fig. 2.) D

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k - 1, the level k is defined as follows. Assume inductively that the submap of M defined by the vertices of levels up to k - 1 is a maximal map Mk-1 and that the vertices of level k - 1 belong with a and b to the external boundary Lk-1 of M k - l , where they are met in a clockwise search from a in the shelling order. Consider as the leftmost vertex of the level k the unlabelled vertex of M such that its neighbors belonging to Lk--lare consecutive on and take the one which has the leftmost sequence on L k - 1 ; label it just after the right vertex of the level k - 1. For the other vertices of a level k, repeat the same search but requiring also that the candidate has no neighbor among the elements already labelled in level k; and label them one by one in the order they are obtained. It is obvious that the above induction is justified since the addition of the vertices of level k to Mk-ldefines a new maximal planar map where these vertices appear on the external boundary in the shelling order. Note that the vertex c is not included in the shelling labelling.

2. The scaffold code for binary graphs

Given a permutation t of [l,n ] , define a sequence of t to be a sequence of consecutive elements, which appear in that order in t, but not necessarily consecutively. A straight of t is a sequence of t such that between any two consecutive elements of it, there exists in t at least one element smaller than both of them, and such that it is maximal for this property. Any permutation on [l,n ] has a unique straight decomposition whose straights are ordered. This can be seen by constructing a decomposition into straights as follows. The first straight consists of the element 1 alone. Subsequence straights are constructed by taking the smallest element not yet included in a straight and constructing the straight of which it is the smallest element. If we number the straight in the order in which they are constructed, the number associated with a straight will be called its index. The index of an element is the index of the straight to which it belongs. The feft nearest smaller neighbor of x is the first element less than x encountered from x on the left. The right nearest smaller neighbor is defines similarly. At least one must exist. If x is an element of [l,n ] , the support of x , s ( x ) , is defined as the nearest smaller neighbor with larger index. It is easily seen that s ( x ) is unique, that is, that the nearest smaller neighbors of x cannot be in the same straight. For let 1 and r be the left and right nearest smaller neighbors of x . If 1 and r are in the same straight, there exists between them an element m less than both of them. This contradicts the definition of 1 and r. We have then the following result. Lemma 1. A permutation t on [l,n ] is a scaffold permutation if and onfy if the support of each element x of index r has index r - 1.

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Associate with the permutation z satisfying the iemma, the binary tree T ( t ) whose vertices are the elements of [l, n ] , with root 1. Whenever the support s ( x ) of x is placed in z on the left (resp. right) of x , define x as the right (resp. left) child of s ( x ) . It is easy to verify that the scaffold permutation of T ( t ) is exactly the permutation t. We have then the following results.

Theorem 1. The set of scaffold permutations on [l,n ] constitutes a code for the set of unlabelled binary trees with n vertices. Now, given a bipermutation u of [l, n ] , define a sequence of u to be a bisequence [a, Q, a + 1, a 1, . . . , b , b ] of consecutive elements, at least two, which appear in that order in u, but not necessarily consecutively as they appear in the sequence. A straight of a consistently nested bipermutation (7 is a sequence of u, such that two different consecutive elements of the sequence have between them in u an element smaller than both of them, and maximal for this property. Any bipermutation on [l,n ] has a unique straight decomposition, whose straights are ordered, the first one of index 1 containing just the element 1, as above. If the two occurrences of x are not consecutive in a, define the left support of x, s , ( x ) , to be the left nearest smaller neighbor of the first occurrence of x ; and the right support of x, s&), to be the right nearest smaller neighbor of the second occurrence of x. If the two occurrences of x are consecutive, the support of x is a single element, s ( x ) , namely whichever of the two elements just defined has larger index (see Fig. 3). We have then the following result.

+

Lemma 2. A bipermutation u on [l, n ] is a scaffold bipermutation if and only $, u is consistently nested, and for each element x of index r there exists in u, between s , ( x ) arid s 2 ( x ) inclusive, an occurrence of an element which has index r - 1. Zf the two occurrences of x are consecutive this requires that s ( x ) has index r - 1. Associate with a bipermutation u satisfying Lemma 2, the binary graph B ( a ) whose vertices are the elements of [l, n ] , and having root 1. Whenever a support F of x is placed in CT o n the left (resp. right) of x , define x as the right (resp. left) child of s. It is easy to verify that the scaffold bipermutation of B ( a ) is then exactly the bipermutation u. We have then the following result.

Theorem 2. The set of scaffold bipermutations on [I, n ] constitutes a code for the set of.udabelled binary graphs with n vertices.

3. The scaffold code for maximal planar maps Given a rooted maximal planar map M with n + 3 vertices, we define its binary graph R ( M ) with n vertices in the following way.

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341

First determine the shelling order of the vertices, and orient each edge from the vertex occurring earlier in this order to the one occurring later. We say that a triangle xyz “hangs” from x if (y, x ) and (z, x ) are oriented towards x . If (y, z) is oriented from y to z, we say that z is the middle vertex of the triangle. The fan of x is defined as the set of triangles hanging from x . The fan x may be a single triangle, but, if there are two or more triangles, they can be ordered from left to right so that each one except the last has an edge incident to x in common with its successor (see Fig. 3). We define vertex 1to be the root of B(M).For any vertex x , other than a, 6 , c and 1, we consider the leftmost and rightmost triangles of the fan of x (they could be the same triangle). The left and right fathers of x are defined to be the middle vertices of these two triangles. If the triangles are the same, then x has a single father. Suppose z is a father of x , as defined by the triangle xyz. Then x is a right son of z if (z, y) is the rightmost edge, incident to z , of the fan of z. Similarly x is a left son of z if (z, y) is the leftmost such edge. It can be verified that the labelling of the vertices of B(M)is exactly its shelling labelling. Conversely, given a binary graph B, consider the shelling labelling and build a maximal planar map M on a triangle (a, b, 1) as follows. Take vertices in the shelling order, and for each vertex x construct the fan of x , which is possible since the fan is determined by the two fathers (or one father) of x , and since it is known whether x is a right son or a left son. The map M has now been constructed except that the vertex c is missing. We add this vertex in the external face and join it to all vertices on the boundary. It can be verified that the planar graph M obtained is such that B(M)= B. Hence we have the following result. Theorem 3. The set of scaffold bipermutations on [l,n ] constitutes a code for the set of rooted maximal planar maps with n + 3 vertices. This follows since we have shown that there is a bijection between the set of scaffold bipermutations and the set of binary graphs, and also a bijection between the latter set and the set of maximal planar maps.

4. Some algorithmic remarks

Given a binary tree, or a binary graph with n vertices, the construction of its scaffold permutation or bipermutation is easily done in O(n) time by a left-first depth-first search; for the binary graph the split of vertices of indegree two is necessary. In the course of the search each vertex is written once at the time of the search of its right child, which may exist or not. On the other hand the recognition of the scaffold property on a permutation or bipermutation seems to need O(n log n) time.

342

P. Rosensfiehl

The table of inversion of a scaffold permutation whose binary tree is known is also computed in O(n) time by the same search as above. In the course of the search, and for each level, the number of written vertices is kept, and a totalizer keeps for each move the number of written vertices of higher level than that of the current vertex of the search. The same algorithm can be developed for a scaffold bipermutation whose binary graph is known. However, the number of steps for handling the totalizer is equal to the sum of the lengths of the edges, where we define the length of an edge as the difference of the levels of its vertices. For this reason it is not clear whether the table of inversion of a scaffold bipermutation can be constructed in linear time. Another approach would be to take advantage of the fact that any scaffold permutation on n elements is contained in a Jordan permutation of 4n elements, and that Jordan permutation can be sorted in linear time [2].

Acknowledgement We would like to thank Ph. Flajolet, H. de Fraysseix and R.C. Read for helpful discussions. This research was supported in part by the “P.R.C. Mathdmatiques et Informatique” of the French government.

References H. de Fraysseix, J . Pach and R. Pollack, Small sets supporting Fkry embeddings of planar graphs, Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing (1988) 426-433. K. Hoffmann, K. Mehlhorn, P. Rosenstiehl and R.E. Tarjan, Sorting Jordan sequences in linear time using level-linked search trees, Information and Control, 68 1-3 (Academic Press, February-March 1986) 170-184. [3] D.E. Knuth, Sorting and Searching, The Art of Computer Programming, Vol. 3 (Addison-Wesley Pub. CO, 1973) 11-13. (41 D.D. Sleator and R.E. Tarjan, Self-adjusting binary trces, Proc. Fifteenth Annual ACM Symp. on Theory of Computing (1983) 235-245.

Discrete Mathematics 75 (1989) 343-372 North-Holland

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BOUNDS ON THE MEASURABLE CHROMATIC NUMBER OF R" "Sight may distinguish of colours; but suddenly To nominate them all, 's impossible." -W. Shakespeare, King Henry VZ, Part ZZ. L.A. SZEKELY* ELTE Matematikai IntpZet, Muzeum Krt 6 8 , Budnpest, 1088 Hungary

N .C . WORMALD Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand

We develop a method of estimating the (upper) density of a set in R" or S" for which the distance between any pair of points is not in a prescribed set. This is a generalisation of a planar principle of the first author. It improves the best known results for small values of n 3 3. It also improves the known lower bounds on the measurable chromatic number of R" for small ns4.

1. Introduction Let H be a set of positive real numbers. We define the geometric graph GHas follows: its vertex set is R" or S", and two vertices are adjacent if the distance between them belongs to H. If H = (1) we write G1. A set A of vertices is H-independent if A is independent in G H . The problem we consider is: how big can an H-independent set be? That is, how big can a set be if certain distances are forbidden between its points? The question of size confines our attention to Lebesgue measurable sets and leads us to the notion of upper Lebesgue density. We define the density threshold, mH, of H-independent sets in S"(r) and R", as follows: m$)(r) = sup {an(A)/an(Sn(r)) :Ac S"(r) measurable H-independent) A

mS;) = ~ $ ) ( o o ) lim sup I(A fl Z)/I(Z):Z cube, A c R" measurable H-independent Here a,,is the area measure in R"+l, Sn(r)is a sphere of radius r in R"+', and A is the Lebesgue measure. If H = { 1) we write m lfor mH. 'This research was carried out when this author was a postdoctoral fellow at the University of Auckland. 0012-365X/89/$3.50@ 1989, Elsevier Science Publishers B.V.(North-Holland)

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The importance of the density threshold originates from its relation to the chromatic number of geometric graphs. The measurable chromatic number x,,, of GH is defined in [34] as the minimum number of colours required in a proper colouring of the vertices of GH having Lebesgue measurable colour classes. The well-known graph theoretic inequality cux2JVI (where a is the size of a maximum independent set, x is the chromatic number, and V is the vertex set) has its counterpart: mHXm 3 1.

This makes it possible to give a lower bound for xm as the reciprocal of the density threshold. Obviously, we also have x 6 x,,,. As observed in [34], the concept of measurable chromatic number is as natural as the concept of chromatic number, and perhaps these numbers differ. On the other hand, if we reject the Axiom of Choice and suppose that all sets are measurable in R", then these concepts are identical. Making such an assumption, we lose the de Bruijn-Erdos theorem [l] which states that for k E N a graph is k-colourable if and only if all its finite subgraphs are. The advantage in investigating x,,, is that good lower bounds can be given by good upper bounds on mH proved by analytic tools. We use the following notation: S"(r): a sphere in Rn+l with radius r ; S n ( m ) is identified with R". S"(z, r ) : an S"(r) with centre z. G: the group of orientation-preserving orthogonal transformations either of S", or of R, and fixing the origin. h: the normalized Haar measure of G. 0 , : the area measure in A: the Lebesgue measure. xe: the characteristic function of the set B. x: chromatic number; x(R"):x(C,) in R". xm: measureable chromatic number; x,,,(R") :x,,,(G,) in R". PQ: the distance between P and Q. P: the vector from the origin to P in R". P: probability. The first systematic study of density thresholds was made by Larman and Rogers [23] who introduced the following configurational principle.

Theorem A. tf there are M points (multiple points permitted) in R"(S"(r))and every ( D + 1)-subset of these points has at least one pair of points determining a distance belonging to H , then

rng) s D / M ( m f ; ) ( rS) D / M ) . Larman and Rogers proved Theorem A only for R", making use of the translation group structure of R". Such a proof cannot work for S", except for

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345

n = 1 and 3 of course. Frankl and Wilson [15] applied it to spheres, but without a detailed proof. We give a proof now since we have some comments to make on the method, and we need the result for spheres.

Proof. Let A c S"(r) Be measurable and H-independent, and let al, . . . , aM denote the M points. By the independence of A, M

for every g E G. As a consequence of Lemma 5 in Section 2, since A+JG XA(g(ai))dh is an isometry-invariant measure on the sphere, we have lGXA(g(ai))dh = an (A)

(s"(r))*

Hence, integration of (1.1) by h yields

The two methods of proof of Theorem A exemplify the two dual approaches of the present paper: sieve or inclusion-exclusion formulas for translated copies of measurable H-independent sets; and polynomials of characteristic functions of sets evaluated pointwise in connexion with a configuration and integrated on an isometry group. The first approach, sifting sets, is not as widely applicable as the second, seemingly not applying to S" for n f 1 or 3, but in R" it provides a less intricate argument than the second, sifting points, would require. To set the scene for our main result, we mention the first author's recent introduction of the following planar configurational principle [34].

Theorem B. Suppose t = inf H E H . Let Pl, . . . , PM be a configuration of M points in the plane. If p = I{(i, j ) :i <j , Sq < t / 2 } 1 ,

q = 1 {(i, j ) :i < j ,

M.3t / 2 , Sq $ H } 1,

then

The main aim of the present paper is to give the following generalisation of Theorem B, and some of its applications.

Theorem 1. Suppose Pl, . . . , PMis a finite sequence of points in S"(r), where

L A . Sz6kely, N.C. Wontnld

346

n >, 1, 0 < r =s00 and multiple points are allowed. Let

with mE-')(O) = 1. Then

where k = [ 2 2 / M 1 .

To see that Theorem B is implied by Theorem 1, we observe that when n = 2 and r = m, m L - ' ( e . ) can be estimated by 4 or 1 according as is at least, or less than, min H , and it is 0 if %.E H. Theroem 1 actually holds for all k 2 1for which krn - 2 > 0; the stated value of k gives the best upper bound (see Lemma 4 in Section 2 ) while Theorem B uses k = 1. In the following section we review the known results related to ours. Section 3 contains some probabilistic and measure-theoretic lemmas necessary for the proof of Theorem 1. We note here that combining Lemmas 3 and 4 gives the following apparently new result.

e.

meorem 2. Let A l , . . . , AM be events of the same probability p and let

2 = p - l Ci,j P(A, nAi). Then p

;

(k l )

6-

kM-X

where k = [ 2 X / M ] . Theorem 1 is proved in Section 4. Our main use of Theorem 1 is to investigate the geometric graph G, and to improve a number of upper bounds for m y ) and lower bounds for xm. This is done by iterating the application of Theorem 1 to a number of appropriate configurations of increasing dimension. Our method is easier to handle in practical computations than the use of Theorem A. This is because the best use of the latter requires knowledge of the maximum size of an independent set in the subgraph of GH induced by the M points, which requires the solution of an NP-complete problem in general. In [23], for some big configurations D could only be estimated by the Davenport-Haj6s Lemma. The configurations we found useful are described in Section 5 . The algorithm we used in applying Theorem 1 to these cofigurations was complicated by the fact that many configurations were involved. We describe our computer implementation of this algorithm in Section 6. The results are presented in Section 7, together with a complete proof of our bounds on m y ) and xm(R") for 3 =sn s 9. Related unsolved problems are given in Section 8.

Bow& on the measureable chromatic number of R"

347

2. History

Investigation in this direction was initiated by Hadwiger [18], who proved in 1944 that if R" is covered by n + 1 closed sets (n 3 l), then one of the sets realises all positive reals as the distance between two of its points. In 1970, Raiskii [30, 311 dropped the restriction "closed" for n 3 2, and Woodall [28] independently did the same in 1973. Generalising Raiskii's idea, Larman and Rogers [23] proved Theorem A, and the following result, in 1972.

Theorem C . Suppose there are M points in R" (multiple points allowed) such that every ( D 1)-subset of these points has at least one pair of points at unit distance apart. Then any covering of R" using less than MID sets has at least one set realising all the positive reals as the distance between two of its points.

+

The existence of a configuration as in Theorem C clearly imples x(Rn) 3 MID, whereas Raiskii's theorem implies x(R") 3 n + 2. The best known bounds in dimensions 2 and 3 are . 4 s x(R2)

6 xm(R2)

7

(Hadwiger [19], L. and W. Moser [27]),

5dxm(R2), 64x,,,(R3)

(Falconer [14]),

5 6 x(R3) c %,,,(R3) s 21

(Raiskii [30], [31]; the colouring is an exercise for the reader).

Frank1 and Wilson [14] demonstrated the exponential nature of %(BY),thereby solving a longstanding problem of P. Erdos, by proving that x(R") 3 (1.2 + o(1))" and m y ' s (1.2 + u(l))-". On the other hand, x,,,(Rn) d (3 + o(1))" and hence m p ) a (3 + o(l))-", as proved by Larman and Rogers [23], using previous results of Butler [2] and Erdos and Rogers [ll]. Falconer [14] recently proved, without using my),that xm(R") 3 n 3, which is much worse than the Frankl-Wilson bound in high dimensions but notable if n = 2 or 3. On the other hand, the Frankl-Wilson bound is weak even in the 25th dimension. The Moser spindle (Fig. 1) and Theorem A prove that mf)s 3, and a natural generalisation, the Moser-Raiskii spindle, proves mi2)d [23]. These estimates

+

Fig. 1. The Moser spindle.

L.A. Szlkely, N.C. W o d

348

were improved by the first author to mf) < [34] (and further [33]) and mi3)s 5 [351. The latter inequality is an alternative proof for xm(R3)2 6. On the other hand, an easy lower bound is mi2)2 n / ( 8 f i )= 0.2267, given by a configuration of open disks of diameter 1 with centres at the vertices of a lattice of equilateral triangles of diameter 2. L. Moser obtained mi2)9 0.2293 by a slight but tricky modification of this set (see Croft [8] and W. Moser [B]). In Table 1 we quote the results of Larman and Rogers [23] for dimensions n up

Table 1. The best known applications of Theorem A to lower bounds MID for the reciprocal of m y ) , n S 25. By Theorem C, M I D is a lower bound for x(R"). n

M

D

2

7

2

Moser spindle

3

14

3

Moser-Raiskii spindle

4

23

4

Moser-Raiskii spindle

5

16

2

Half-cube

6

316

32

7

8

56 64

4

7-dimensional Gosset polytope Special Gosset spindle

9

64

4

Special Gosset spindle

10 11

165 220

9 12

18: 18

12

286 364

12 12

23

13 14

455

13

35

15

560 680

16 16

35 42;

17

816

16

51

18 19

969 1140

17

20

57 57

20 21

1330

20

22

1540 1771

20 21

23 24

4600 4692

46 46

25

2600

24

16

Configuration

Half-cube spindle

4

'

30

66; 77 84;

>

Erdos-T-S6s configurations

, Leech-Conway configuration Special Leech-Conway spindle

J

Erdos-T-S6s configuration "

Boundr on the measureable chromatic number of R"

349

to 25. Of these, only when n = 2 and 3 were better bounds for m y ) obtained prior to our present results.

3. Prelimianrylemmas The first two lemmas, due to RCnyi, are equivalent. We will use the second explicitly, and we include the first for comparison with a step in the proof of Theorem 1.

Lemma 1 (Rdnyi). Let f be any function from (0, l}"to R , and Q a probability space with measure o.Then I f (xAI,. . . ,xAn)d o 3 o for all events A ~. ., . ,A , c 8

iff

fkA,, . . . ,xA,)(w)2 0

for all sequences A l , . . . ,A ,

of events of probability 0 or 1 in Q and all w E R Lemma 2 (Rdnyi [26, 321). For i = 1,2, . . . ,k let Bi be a Boolean polynomial (with operations U,

f l

and complementation) and let ci be a constant. Then

2 ciP(Bi(Al,. . . ,A,)) 2 0 i

for all sequences A l , . . . ,A , of events in a probability space 8 if and only if the inequality holdr for all sequences A l , . . . ,A , of events in 8 of probabiZity 0 or 1. RCnyi actually proved only Lemma 2, but the equivalence with Lemma 1 is easy to see.

Lemma 3 (Kai Lai Chung [22]). If A l , . . . , A , are any events in a probability space and k is a natural number, then

Proof. By Lemma 2, it is enough to check that the inequality is true whenever exactly t events have probability 1 and the rest have probability 0 (t = 0, . . . , n). In this case, the inequality is equivalent to (k - t)2+ (k - t ) 3 0. 0 Lemma 3 can be strengthened, as is plain from its proof, but as we shall see, it is optimal for our present purposes. Its bound is optimized in turn by the following lemma, which is straightforward to check.

350

L.A. Sz&dy,

N.C. Wormald

Lemma4. LetM>O, z T > O , p > O . If

pSMk-Z

for every natural number k > . Z / M , then the sharpest of these bounds is for k = [2.Z/M]. It may be surprising that, although the method proof of Lemma 3 can give a more general result, we only require the stated version. This is because the way we shall use Lemmas 3 and 4 is the following. For a certain natural number M, and a fixed 2, reals a and b satisfying a 3 bt - (i) for all t = 0, . . . , M and 6 > Z / M will lead to the inequality m g ) ( r )s a / ( b m- 2).Thus, the sharpest inequality occurs when (b/a)M - (l/a).Z is maximised, subject to 1 3 ( b / a ) t( l / a ) ( ; ) for t = 0 , . . . ,M . In this linear programming problem, the maximum occurs when at least two of the constraints hold simultaneously, and it is easily seen that these must be consecutive integers t. It follows that a = ( " t ' )and b is integral. The next result apparently follows immediately from results in the theory of relative invariant measures in homogeneous spaces (see Helgason [21]). We prove it here since its extraction from that theory is difficult, and the proof, while using techniques reminiscent of that theory, is quite simple.

Lemma 5 . Every finite isometry-invariant measure on the Lebesgue-measurable sets of S" is confor some nonnegative constant c. Proof. The area a,, is isometry-invariant. Suppose p is an isometry-invariant measure on S" such that a,(S")=p(S") and there exists a measurable set X c S" with

Since both of

a,

and p are isometry-invariant, we have for every g E G

and hence

On the other hand, the function z E S"+

JG

k x ( g ( z ) )dh is constant and by

Boundr on the measureable chromatic number of W"

p(S") =

351

we have

We now have a contradiction by (3.1), (3.2) and Fubini's theorem. 0 It is also possible to give a rather elementary proof of Lemma 5 for n = 2 by the following sequence of steps. If p is isometry-invariant then p = can on: the digons determined by great circles (lunes) having an angle ( p / q ) n ,all lunes, all spheric triangles (by Girard's area formula using the areas of lunes), all open sets. For n = 1 and 3, the lemma also follows immediately from the unicity of the Haar measure, as these spheres are topological groups. For z E S"(r) and p > 0, let T ( z ,p) denote S"(r) n S"(Z, p ) .

Proof. Let G' denote the subgroup of G fixing P, with corresponding measure h', and let g' denote an element of G'. We prove the lemma by showing Zl(P, Q) = JG, Zl(P, g ' ( Q ) ) dh' = J G ~Z2(P, g'(Q)) dh' = Z2(P, Q). The third of these is immediate, and the first follows from (g')-' E G' c G, so we examine the second in detail. We have

(by Fubini's theorem),

(by Lemma 5, regarding the second integral as the measure of X t l T ( g ( P ) ,

(by the invariance of the Haar measure on G), = Z2(P,

Q)

(after reversing the order of the first two integrations). 0

m)),

352

L.A. SzAely, N.C. Wonnald

4. Roof of Tbeorem 1

Let X be an H-independent measurable set in S"(r). We assume without loss of generality that X is closed, since in any case X contains a closed set whose measure (or upper Lebesgue density, if r = = ) differs from that of X by a negligible amount. Suppose firstly that S"(r) is itself a topological group; i.e. r = or r < m and n = 1 or 3. Let C be a big cube if r = TX: and let C = S"(r) if r < w. Let * denote the group operation; i.e. addition in R", multiplication of complex numbers of unit absolute value in S'(r) (with r scaled to 1) and multiplication of unit quaternions in S3(r). Applying Lemma 3 to A, = P, * (Xn C) in the probability spaces

and P ( A i )= an ( A ' ) in

S" (for arbitrary n )

(4.1)'

Un(S")

(4.2') In (4.2), the error term o(1) is for A(C)+=. After slight modification, we get

(4.3') We next integrate the inequalities (4.3) and (4.3') over g E G where the positions of PI, . . . , PM have been moved by g. The only non-constant terms are the numerators in the terms in the summations. They become

and

Bounds on the measureable chromatic number of R"

353

where we have used for brevity the notations

Edz) = and

E2(Z) =

I I

X-g(Pi)+g(4)+x"c(z)

G

Xg(P,)-l.g(P,)*X(Z)

G

dh

dh.

e.,

For fixed 4, z E R" (or S") and C, and an arbitrary set X,the values of E&) and E2(z) depend only on the intersection of X with the spheres S"(Z, and S" n S"(Z, respectively, and it is easy to check that they are normalised isometry-invariant measures on these spheres. Hence, by Lemnia 5,

q)

2.)

and

z) is identically equal We also have that xxnc(z)El(z) (and ~ ~ ( z ) E , (respectively) to 0 if E H. Hence integration of (3.3) and (3.3') over g E G yields

( ')

- Z)A(X n C)lA(C) + o(1)

3 (Mk

(4.4)

and

(k

') (Mk- Zbn(X)/an(S"). 3

(4.4')

The theorem now follows for Iw" and S' and S3,and the stated value of k is optimal by Lemma 4. To complete the proof, we consider arbitrary Sn(r) with r c c o . Let b l , . . . , bM E Y ( r ) and let k be a natural number. Then

(In fact, integration of this inequality gives an alternative proof for Lemma 3, and integration of a more general one gives Lemma 2.) Setting bi = g(e.) in (3.5) and integrating over g E G, we get

By Lemma 5 , the first integral is a,(X)/a,(S"). The second integral is clearly 0 if -

ce E H,and otherwise, by Lemma 6, it has the upper bound mg-')(%Vl

-(qq/2i)2~"(x)/fz,(S").

The theorem now follows. 0

L.A. SzCkely, N.C. Wormald

354

We remark that the proof given for S"(r) (sifting points) works for R" as well if the integration of (4.5) to (4.6) is on an appropriate big part of the isometry group of R" of type G x C rather than on G,but the formulae become more complicated. Although we could not apply the argument given for groups (sifting sets) to non-group spheres, the dual approaches of sifting points and sifting sets are in accordance with the duality of Lemmas 1 and 2. 5. List of configurations

Here we list and briefly describe the configurations that we need. The dimension of the configuration is denoted by n, M is the number of points, r is the radius and x +-k denotes that the distance fi occurs k times. S ( k , rn, n) denotes a Steiner system with these parameters. Existence and properties of the Steiner systems used can be found in [3] and [20]. Some of these configurations are spherical designs or subsets thereof, in the sense of [9]. nZcube Take n coordinates, put f 1 into two and 0 into n - 2 of them in every possible way. Here 3 S n c 24, M = 2(n2 - n), r2 = 2, 4+-48(:)+4(;), 6+-8n( n - 1 8tn(n-1). 2+8n( n - 1

),

),

n3-cube Take n coordinates, put f l into three and 0 into n - 2 of them in every possible way. Here 3 C n s 24,

M

= 8(

6+-640(:)

;)

, r2 = 3, 2 +96(:),

+ 192(:),

8+96(n

4 +%(n

4

')n

+ 12(;),

4

l>n

+ 12(4),

10+-96(:),

12+4(;).

Simplex Take a regular simplex of unit edge length. Here 3 s n ~ 2 4 M , =n Let t2 = f , t, = 1 - ((n - l)/n)2fn-,; then r: = (n/(n + 1))2tn. 1+-(" l').

+ 1,

Cross polytopes See [5]. Here 3 s n S 24, M = 2n, r2 = $, 1+- (7)- n, 2 e n .

Erd6s- T. Sbs configuration See [23]. Take n + 1 coordinates, put 1 into three and 0 into n - 2 of them in every possible way. Here 10s n S 24, M = (" 3 I ) , r2 = 3(n2 - n - 2)/(n + 1)2,

')(" 2

;l>o 52)/2.

'), 4-(;)(" i 2 ) ( n + 1)/2, 6 2+('; (In some dimensions these configurations are the best for the Larman-Rogers principle.)

355

Bounds on the measureable chromatic number of R"

Steiner-3

Take a Steiner system S(2, 3, n) [20] and let the vertices of the conguration be the columns of the incidence matrix of S(2, 3, n) (i.e. the characteristic functions of the edges). Here n - 1= 6, 8, 12, 14, 18, 20, 24; M = 4(2), r2 = 3(n - 3)/n,

The dimension of the configuration is n - 1, since the sum of the coordinates of vertices is constant. Steiner-3-cube

Sign the nonzero entries in the vertices of Steiner-3 in all possible ways. Here n =7, 9, 13, 15, 19, 21; M =4(n - l)n/3, r 2 = 3, 4 t (4n - 10)n(n - l), 6 t 8 +(4n - lO)n(n - l),

-y-

$;)(:(I)

1)

12 +2(n - l)n/3.

We remark that the Steiner-3-cube for n = 7 is the 7-dimensional Gosset polytope; see [36]. Steiner-4

Take a Steiner system S(3, 4, n) [20] and let the vertices of the configuration be the columns of the incidence matrix of S(3,4, n). Here n - 1= 7, 9, 13, 15, 19, 21; M = i(;), r2 = 2(n - 4)/n, 4+i(;)(n-4),

6+6(1)(4( n - 1 )-6n+20),

8+:(1)(!(1)-$("2')+3n-9).

Steiner-4-cube

Sign the non zero entries in the vertices of Steiner-4 in all possible ways. Here n = 8, 10, 14, 16, 20, 22; M = 4(3, r2 = 4, 4+(24n-88)(1), 8+32(9)(:(1)

6+64(1)(

(n - l)(n - 2)

- $(n - l)(n -2) +3n -9) - - +25 ) ,

3n

- - +25 )

+ 48(1)(n

'

- 4) + 12(;),

12+-(24n -88)(J),

L.A. Sz&kcly, N.C. Wormald

356

Steiner 4-318 cube Keep only those vertices of the Steiner-4-cube whose sum of coordinates is zero. Here n - 1 = 7, 9, 13, 15, 19, 21; M = $(:), r2= 4,

4+9(;)(n

- 4),

6-:(;)(24("

8 ~ ; ( 4 ) ( 9 ( ; ) -48( n - 1

2') -

108n + 360),

) + 156n -492),

.lOtftjJ)(24( n - 1 )-108n+360),

1 2 ~ (15 ~ ) (n ~ ) ( n - 4 ) ,

Sreiner 3-618 cube Keep only those vertices of the Steiner-3-cubewhose sum of coordinates is f l . Here n = 7 , 9, 13, 15, 19, 21; M =2(;), r 2 = 3 ,

4+(:)(4.5n

- 11.5), 6 t (I)(.(.

8 t n ( n - l), 12

-(; )

- 1) - 4.5n

+ 7.5),

.

Halfcube Consider those 0 - 1 sequences of length n whose sum is even. Here 3 ~ n S 1 3 M=2n-1, , r2 =n / 4 ,

Gosset 6 See 2171, IS]. Here n = 6, M = 27, r2= y , 8 t 2 1 6 , 1 6 ~ 1 3 5 See . [6] for the analogous Hessian polytope. Gosset 8

See [17], IS]. Here n = 8 , M=240, r 2 = 3 , 3 ~ 6 7 2 0 ,6 ~ 1 5 1 2 0 ,9 ~ 6 7 2 0 , 1 2 t 1 2 0 . See [6] for the analogous Witting polytope. The coordinates we used for Gosset 8 are given in [7].

Bounds on the measureable chromatic number of R“

357

Gosset 7 See Steiner-3-cube in 7 dimensions. Pentagonal configuration Represent R 4 as ( x , y, u, v). Put a regular pentagon of unit edgelength centred at the origin into the plane (x, y, 0,O) and put a regular star-pentagon of unit edgelength centred at the origin into the plane (0, 0, u, v). Here n = 4, M = 10, the configuration is not spherical, 16 3 5 , (l6 1)2/4t 5 , (l6- 1)2/4t 5 .

+

Dodecahedron, icosahedron Turned out to be useless. “Other” polytope See [6]. Here n = 6, M = 72, r2= 3, 3 t 7 2 0 , 6 +1080, 9 t 7 2 0 , 12t36. S(5, 8924)

See [3]. Here n = 23, M = 759, r2= 16/3, 8t170016, 12t106260, 1 6 t 11385. S(5, 6 2 4 )

See [3]. Here n =23, M = 7084, r2=4.5, 2 ~ 4 7 8 1 7 0 , 6-2975280,

8 t

8607060, 10 t9563400, 12+3464076.

$66 , W See [3]. Here n = 11, M = 132, r2= 5, 1 ~ 2 9 7 04, t2640, 2 t2970, 3 t66. S(4,5,11) See[3].Heren=10, M=66,rZ=15/22, 1 + 9 9 0 , $ t 6 6 0 , 2 t 4 9 5 . Perm (+ + - - -) Take the permutations of 2 “1” and 3 “-1”. Here n = 4, M = 10, rz = 24/5, 8 ~ 3 0 1, 6 ~ 1 5 . Perm (-11OOO) Permute -110o0 every possible way. Here n = 4, M = 20, 3 = 2, 2 6 6 0 , 4t60, 6 4 4 0 , 8 ~ 1 2 .

L.A. SzCkely, N.C. Wormald

358

Quartercube

-

See [23]. Here n = 10, M = 28, r2 = 5 , 4-2560, 4 3200.

8+14080,

124-12800,

S(4,5,23)

See [3]. Here n =22, M = 1771, r 2 = 90/23, 4679965, 64-371910, 8 t 717255, 104-398475. S ( 4,7,W

See [3]. Here n = 22, M = 253, r2 = 112/23, 6 t 17710, 12t 14168. The following few configurations are direct products. The vertices of the product configuration are ( u l , .. . , v,, ul,.. . ,urn), where ( u l , . . . , u,) is a vertex of the first and (ul, . . . ,u,) is a vertex of the second'configuration. Si denotes the regular unit simplex of R'.

Gosset 8 x S1 n = 9 , M=480,r2=3.75, 3-13680, 240.

6+43680, 9-43680,

12-13680,

15t

n = 10, M = 720, r2 = 4, 3 ~ 2 0 8 8 0 64-85680, , 9t110880, 12-40680,

15t

Cosset 8 x S2 720.

Gosset 8 x S3

n = 11, M =960, ? =4.125, 3t28320, 64-141120, 9t208320, 124-81120, 15+- 1440. Cosset 6 x Cosset 6 n = 12,

M =729, ? = 4 , 3+-11664, 6+-100602, 9+-116640,

12-36450.

Gosset 6 x Cosset 8 n = 14, M = 6480, r2= 5, 3 4-233280, 6~3343680,9 4-8527680, 12 4-6988680, 15t 1866240, 18+- 32400.

Cosset 6 x Cosset 7 n = 13, M = 1512, r2 = 4.25, 3 t32508, 6-354564, 15c 7560.

9 4 3 1 4 6 8 , 12 -216216,

Boundr on the measureable chromatic number of R"

359

Gosset 7 X Gosset 7

n = 14, M = 3136, r2= 4.5, 3 ~ 8 4 6 7 2 , 6 + 1227744, 9 ~ 2 2 8 9 2 8 0 , 12 c 1227744, 15 t84672, 18 t 1568. Moser configuration

See Fig. 1. This configuration is realizable on S2(r) if 0.5862 < r < 0.6277 or 0.8195 < r. Morning star

Take a regular unit simplex centred at the origin in R3. Reflect all the vertices in the opposite faces and keep two copies of the vertices of the original simplex. Here n = 3, M = 12, it is non-spherical, O t 4 , 1 c48,1 2 t 8 , v ' c 6 , where 1 is the distance of a vertex from its mirror image in the opposite face, and v is the distance between two outer vertices. The morning star can be made into a spherical configuration in R4by translating the outer vertices by a distance i along the fourth dimension axis, keeping the unit distances. If x is the distance from one of the outer vertices to the vertex of the opposite face of the original tehahedron in R3, r is the radius of the S3containing the configuration and j is the fourth coordinate of its centre, then the following relations hold: x2+i2=1

rZ = ( j - i)2 +

(m + id)'

r'=j'+:

P = i2 + v2 = !

(a+ V x i Q '

( V m + id)'.

We often use a subconfiguration W of a spherical configuration T. A useful tactic is to specify one point in T as the pole, and then let W = T f l S where S is a sphere of radius q centered at the pole. If q is the minimum distance occurring in T then W is the arctic. If q is lh times the radius of T then W is the equator. G2

The equator of Gosset 8 choosing any of its points as a pole. Here n = 7, M = 126, ?=8, 8 t 2 0 1 6 , 16+3780, 2 4 ~ 2 0 1 6 3, 2 t 6 3 . G21

The intersection of the equator of Gosset 8 with pole (1, 1, 1, 1, 1, 1, 1 , l ) and the arctic of Gosset 8 with pole (0, 0, 0, 0 , 2 , 0 , 0 , 2 ) . Here n = 6, M = 32, r2 = 6, 84-240, 1 6 t 2 4 0 , 2 4 t 1 6 .

360

L A . Sz&kely, N.C. Womrold

G22 The intersection of the equators of Gosset 8 for (1,1,1,1,1,1,1,1) and (0,0,0,0,2,0,0,2). Here n = 6 , M=60, ?=8, 8 ~ 4 8 0 ,1 6 t 7 8 0 , 24t480, 3 2 ~ 3 0(We . note that the arctic of the Gosset 8 is the Gosset 7.)

Limpace The Steiner system S(5,8,24) generates a set algebra. We suppose the underlying set to be {1,2, . . . ,24}. The vertices of linspace are the characteristic functions of the elements of the set algebra. The blocks of S(5,8,24) (octads) were produced as the orbit of {1,2,3,4,5,8,11,13} under the permutation group Mathiu 24 (see [3]) generated by the three permutations (1,2, . . . ,24); (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15);

(1, 22)(2,11)(3,15)(4,17)(5, 9)(6,19)(7,13) (8,20)(10,16)(12,21)(14,18)(23,24).

A n y arctic of linspace is isometric to S(5,8,24) and any equator is isometric to the set of 12-element sets in the algebra (dodecads). We kept the dodecads as increasing sequences of 12 elements of { 1,2, . . . ,24}, lexicographically ordered.

a, a.

Only four different distances occur in linspace; fi, 4 and The notation doili2- ik means the subconfiguration of doili2 - - - ik-l lying at distance fi from the first point of ilj2 * * * i k V l , while dodecs = do is the whole set of dodecads. A similar notation is used for octads. All these configurations are kept in lexicographical order. The statistics relating to these configurations are given in Table 2. Here, the column headed by YIN answers the question of whether this configuration was actually used in proving the bounds on rnp) and rn$)(r) given

--

Table 2. Subconfigurationsof the dodecads and octads. Y/N

N Y N Y N N N Y

N Y N N N

Configuration do do2 do3 do22 do23 do32 do33 do332 do333 do3332 do3333 oct2 oct3

n 23 22 22 21 21 21 21 20 20 19 19 22 22

r 46 4/43 46 45 421/2 4/43 46 4/43 d6 4/43 46 45 421/2

n 2576 495 1584 184 288 315 952

195 560 117 324 280 448

d2

d3

d4

637560 45540 249480 8408 16704 18540 93060 7080 32760 2520 11250 19600 40320

2040192 71280 153904 8208 23040 28440 266080 10800 90720 3852 29664 19040 56448

637560 5445 249480 220 1584 2475 93060 1035 32760 414 11250 420 3360

d6 1288 0 792 0 0

0 476 0 280 0 162 0 0

Bounds on the measureable chromatic number of R"

361

in Table 6. (Y = Yes.) The column headed by di refers to the frequency of the distance fi.Note also that the table gives r2, which is rational.

L The Leech lattice was discovered by Leech [24,25] and many of its remarkable properties were found by Conway [4]. We use Conway's description of it, as a set of lattice points on the sphere of radius 4~ centred at the origin. We call this configuration L. The squares of distances occumng in L are 16i (i = 2,3, 4, 5,6, 8). Specifying a linear order of the vertices of L, we define subconfigurations Lil - is - . ik (is E {2,3,4, 5, 6, 8}) in the same way as we did for dodecs, with the one difference that the pole of L used in generating L2, L3 and L4 was (-4, - 4 , 0 , . . . ,0) rather than the first point of L in the linear ordering. For all other configurations, the pole was the first point stored, and points in a subconfiguration are stored in the same order as in L. To specify the linear order of the vertices of L (we did not use lexicographic), we give the following pseudo-pascal program for generating them. As before, each octad is an 8-subset of (1, . . . ,24}, stored as an increasing 8-tuple. Each dodec is a 12-subset of (1, . . . ,24}. Both are read lexicographically. Also, append denotes appending (x(l), . . . ,x(24)) to the output file.

- -

procedure subsl(k, I ) ;

fort:= 1 to 24 tox(t):=O; x ( i ) := k and x ( j ) := 1.

procedure subs2(i, A);

f o r j : = l to 2 4 s e t x ( j ) : = l i f j ~ AanLx(j):=-l otherwise. x(i):= - 3 * x ( i ) . program generate L;

begin for i : = l to 23 do for j : = i

+ 1 to 24 do

begin subsl(4,4); append; subsl(-4, -4); append; subsl(4, -4); append; subsl(-4, -4); append;

end; repeat A:=next octad=a(l),

for k : = O to 127 do

. . . , a(8);

L.A. SzCkely, N.C. Wormald

362

besin (c(l), c(2), . . . 47)) :=the binary representation of k; c(8) : = 0 or 1 so that 2 ( i ) is even; foci:= t o 8 d o x ( a ( i ) ):= 2 if c ( i ) = 0 and -2 otherwise; set x ( i ) : = O for all i $ A ; append; end; for i : = l to 24 do be&

subs2(il A); append ; for j := 1 to 24 do x ( j ) := - x ( j ) ; append; end; until A = last octad;

reperrt A := next dodec; for i:= 1 to 24 do begin subs2(i, A);

append; end; ~ t iAl = last dodec;

for i:= 1 to 24 do begin subs2(i, { 1, 2, . . . , 24));

append; subs2(i10); append; end;

end.

The configurations L, L2, L3, L4, L22, L23, L24, L232 and L233 all have a nice distance-invariance property: the frequencies of the distances from a point x

Table 3. Subconfigurations of L. ~

Y/N Y

N N N Y Y

N

N Y

N Y

N Y Y Y

N Y Y

N N

N N

N N

N N Y

N Y Y

N N

N

N Y

N N Y Y Y

N N N

Configuration

n

L2 23 L4 L22 L23 L3 2 L33 L34 L35 L43 L232 L233 L323 L332 L333 L334 L335 L342 L343 L344 L433 L434 L442 L443 L444 L2333 L3443 L3444 L3445 L4442 L4444 L23333 134434 L34444 L44443 L44444 L3 44443 L344444 L444444 L4444443 L444 44 44 L4444444 4 L4444444444

23 23 23 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 20 20 20 20 20 20 19 19 19 19 19 18 18 18 17 17 16 14

r2

24 30 32 64/3 24 352/15 144/5 448/15 80/3 30 64/3 24 258/11 208/9 28 256/9 220/9 164/7 201/7 208/7 144/5 448/15 24 30 32 24 372/13 384/13 340/13 24 32 24 2624/93 88/3 30 32 312/11 320/11 32 30 32 32 32

M 4600 47104 93150 891 2816 2025 15400 22275 7128 22528 567 1680 1232 972 5892 6885 1620 1092 7392 10290 7336 10431 1360 10240 19962 976 3360 4745 1504 1280 17400 552 1447 2180 4224 7590 684 1003 6372 1536 2658 2144 748

d2 2049300 47692800 114760800 149688 798336 467775 7484400 12162150 2673000 12063744 61236 294840 179256 120285 143256 6 1766610 245430 141330 1917216 2962050 1852200 2942100 195480 2810880 6870240 103032 450624 737532 154528 178560 5795040 34044 98963 186792 581184 1262976 25265 47678 998784 94976 203136 150912 21936

d3

d4

2049300 6476800 524620800 362700800 1049241600 2010363300 18711 228096 798336 2365440 334125 1247400 53014500 45368400 114604875 82328400 9355500 12474000 117494784 82632704 8505 90720 294840 819840 449680 129360 282366 69255 7275420 7231770 10213290 9371700 306180 752490 100380 353976 11946480 10444000 23868705 17821440 11812500 10177440 24584805 17947440 195480 532480 16916480 23731200 88159320 48660480 103032 269248 2410896 2143968 4956372 3821760 386096 555360 178560 460800 65307240 37232640 34044 83712 430848 409622 1029894 800112 3872448 2823168 12163884 7053312 96761 84816 211199 168488 8343972 4976640 493824 358400 1448288 837632 929440 532480 124828 49152

d5 0 167878656 1049241600 0 0 0 12474000 37422000 898128 39739392 0 0 0 0 1400490 2308500 7290 0 2946720 7929600 2993760 8502480 0 8478720 48660480 0 622176 1661760 34272 0 37232640 0 105100 343440 1520640 7053312 26240 71352 4976640 206848 837632 532480 49152

d6 2300 6476800 114760800 0 1408 0 231000 1559250 0 1813504 0 840 0 0 14640 38070 0 0 62720 355110 68880 420840 680 486400 6870240 488 15456 77716 0 640 5795040 276 1648 14872 121536 1262976 504 3786 998784 24832 203136 150912 27936

d8 0 0 46575 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9981 0 0 0 0 0 8700 0 0 0 0 3795 0 0 3186 0 1329 1072 374

5

a

2a. 0

a

H

Q-

3

W

8

L.A. Szkkxly, N.C. Wormald

364

to all other points is independent of x (this follows from the results of [9]). This made it easier to compute their distance distribution (see also [16, p. 1291). The others were done by computing the distances between all pairs of points, using a computer. The results are in Table 3, where Y / N is as for Table 2, and di refers to the frequency of the distance Although we almost always used the first point in a configuration as the pole in generating subconfigurations, other subconfigurations were sometimes found using different poles. However, a little investigation made it appear that no particular advantage was to be gained by using these other configurations.

a.

6. Method of cornputation

The basic items of data computed were ordered pairs, called jump points, of two types: singular and nonsingular. These are arranged into sets Sing(n) and Nonsing(n), 2 =sn S 24. The only requirement is that if a jump point (r, b) is in Sing(n) then m$")(r)d b, and if (r, b) is in Nonsing(n) then mp)(r') s b for all r' r , including r' = 30. Given Sing(k - 1) and Nonsing(k - l), we obtain sets Sing(k) and Nonsing(k) by applying Theorem 1 for each of the configurations which can be embedded in an Sk, or in R k . Theorem A was also applied, but turned out to be useless for k 2 3. Almost all the configurations used are spherical. Such a configuration of dimension n and radius ro gives rise to a jump point (ro,b) in Sing@) if n = k 1, and to a number of jump points (r, b)~Nonsing(k), with r Z r o , if risk. Non-spherical configurations (such as the pentagonal configuration) give rise to jump points with r = m. A convenient place to begin the computation is with dimension 2. Firstly, and trivially, we can put (0, 1) in Nonsing(2). W e also obtain the following elements of Nonsing(2) by applying the Larman-Rogers principle (Theorem A) to the given configurations:

+

(4,

&two

(&,$)-the (0.819417678, +)-the

(G, a)

points at distance 1 apart; vertices of a unit equilateral triangle; Moser configuration.

Finally, put into Sing(2) by applying Theorem A to the vertices of a unit 3-simp1ex in s2(fl). The Moser configuration can also be embedded in S2(r) for 0 . 5 8 6 2 s r s 0.6277. This does not produce any more elements of Nonsing(2), but can be viewed as giving an infinite subset of Sing(2) if required. However, our implementation of the computations made such information difficult to deal with, and a little experimentation showed that it made very little difference to jump

Bounds on the measureable chromatic number of W"

365

points obtained in higher dimensions, and no observable difference to our bound on my)(..) for small n. Hence, this extra interval of existence of an embedding of the Moser configuration was not made use of. (In fact, our vast experience at repeated computations showed that omitting a jump point, or even any single configuration, from the computations, rarely made much difference to the bounds obtained, especially in dimensions much higher than that of the omitted configuration.) Given a list of elements of Sing(k - 1) and Nonsing(k - l), we proceed to calculate elements of Sing(k) and Nonsing(k) as folows, via Theorem 1. In these d lead to a sphere of radius calculations, two points & and 4 with %.= r , = d d m . We call r, the radius induced by d in Sk(r). To estimate mik-')(r1),we use the best upper bound implied by Sing(k - 1) and Nonsing(k 1). The configurations of the correct dimensions are taken one at a time. Except for the morning star, which we postpone discussing, each distance present in the configuration is taken in turn, and the configuration is expanded or contracted so that the chosen distance becomes 1. Its adjusted radius and distances are computed. For a non-spherical configuration, its dimension must be at least k. Theorem 1 is then applied with r = m. The result is an element (00, b) of Sing@). On the other hand, for a spherical configuration, two things are done. Firstly, if its dimension is k + 1, then a singular jump point (ro, b) is obtained by Theorem 1, where ro is the adjusted radius of the configuration. Secondly, for each (adjusted) distance d in the configuration, other than 1, and for each jump point (r, b) in Sing(k - 1) or Nonsing(k - l), do the following. Set rl = d 2 / 2 1 / m (if it is real or infinite). Then embedding the configuration in Sk(rl) and applying Theorem 1 causes mtk-')(r)to be called for when d. Hence, we obtain a jump point (rl, b,). We call this the jump point induced by d and (r, 6 ) . It is singular either if (r, 6) was singular or if any elements of Sing(k - 1) were used for obtaining bounds, and is then added to Sing(k). If (r, b) is in Nonsing(k - l), the computation is repeated using only bounds obtained from Nonsing(k - l), to obtain an element of Nonsing(k). The treatment of the morning star is slightly different, since the distances of the configuration, when embedded in a sphere of given radius, mu$t be computed by a different method. We did this using the equations relating r, 1 and v given in the description in Section 5 . As when treating the other configurations, for each jump point (ro, b) in Sing(k- 1) or Nonsing(k-l), the values of r, I and u were computed so that mik-')(ro) was called for when apply Theorem 1, first with &P, = I, then with = v. If no singular jump points were used for bounds, this gave an element of Nonsing(k), because of the monotonicity of the appropriate functions (we omit details). Otherwise, it gave an element of Sing(k), as long as r, 1 and v were all positive reals. We did not expand or contract the morning star. Finally, all the elements of Sing@ - 1) and Nonsing(k - 1) were directly copied

sq=

=.

366

L.A. Sz&kely, N.C. Wormald

into Sing(k) and Nonsing(k), since they were all obtained by Theorem 1 or Theorem A, and both of these give the same bound when the dimension of the space is increased by any positive integer. This method was implemented in a Pascal program. For a jump point (r, b), the bound b was represented as a real, and we may expect very high accuracy in the results here: probably to 14 significant figures as far as our present computations are concerned. In preliminary computations, we stored r as a real for all jump points, and assumed that b was an upper bound on mik)(r')whenever Ir' - rl < whether or not the jump point was singular. This possibly gave invalid results, erring on the side of providing stronger bounds, and was done because of the inaccuracies in representing reals on a computer. In all later computations, valid bounds were obtained by effectively storing r2 as a rational whenever it had been computed that Nor2 is an integer, where No is any suitable integer. Such a jump point is called precise. With No = 64 x 81 x 5 X 7 x 11, we obtained virtually the same results as with the preliminary computations. For imprecise jump points, r was stored as a real. If a jump point (r, b , ) is induced by a distance d and an imprecise jump point (I, b), then (r, b) is always applicable in the computation of b l , but r, will not be precise. In all other cases that an imprecise jump point (r, b) in Nonsing(k) was used in a computation, it was only applied to mp)(r') for r' > r E, where we used E = One can show that the errors in our computations of the values of r and r' could not exceed ~ / 2 Hence, . the actual value of r is indeed less than r', and so this jump point really does provide a bound on mik)(r').At some other places in the computation, it was also necessary to err on the side of safety. Steps had to be taken to reduce the proliferation of jump points. Obviously, if (rl, b,) is in Nonsing(k) and (r,, b,) is in Sing@) or Nonsing(k) with r, s r2 and b , 6 b Z ,then (r,, 6 2 ) can be deleted. Similarly, if these jump points are both in Sing(k) and r, = r2 and b , zs b2, then delete (r2,b,). This policy still left too many jump points: over lo00 when k - 9 , with the number doubling to tripling with each extra dimension. The extra information carried along with jump points, given necessary details of how they were obtained, limited the total number of jump points to 20,000 or so, when practical considerations were made. To circumvent this problem, note that deleting jump points still gives valid results. A constant 6 was chosen, and at the end of the computation of Sing(k) and Nonsing(k) described above, it was ensured that for every nonsingular jump point (r, b ) retained, all jump points (r', b') with r' > r and b ' a b ( 1 - 6) were deleted. With 8 = 0.001, several thousand jump points were required for all ks24. Reducing 6 below this value did not seem to change any results much. For example, 6 = O.OOO1 gives exactly the same bound on

+

7. Results

The method of computation described in Section 6 was implemented in a Pascal program and run on an IBM 4341 at the University of Auckland. The results

Table 4. Jump points (r, b ) used in the proof of lower bounds on rnp)(r) for n 6 8 and large r. r

jump

sing

r2

O 1/4 3/10 1/3 3/8

3-1 3-2 3-3 3-4 3-5 3-6

0.0 0.500000 0.547722 0.577350 0.612372 0.918417

N N Y N

4-1 4-2 4-3 4-4 4-5 4-6

0.0

N Y

5-1 5-2 5-3 5-4 5-5

0.559016 0.577350 0.784464 0.929876

Y

N

Y N

?

O 5/16 1/3 ?

N

?

Y

-

0.0

N

0.577350 0.612372 0.849836 0.938510

Y

? 1/3 3/8

N

7

6-1 0.0 6-2 0.612372 6-3 .0.884651 6-4 0.945249

N Y N N

O 3/8

7-1 7-2 7-3

0.707106 1.007326 1.033499

Y N N

1/2

8-1

1.079198

N

7

Y

-

o

Y N

?

?

?

b

configuration

dim

unit

1.000000 0.500000 0.400000 0.333333 0.250000 0.187500

copied copied perm2+3 copied simplex morning star

4

2

3 3

1

1.000000 0.312500 0.316216 0.142857 0.137931 0.128000

copied halfcube n2cuba cross polytopa cross polytopa pentagonal

5 5 4 4 4

1.000000 0.222222 0.209773 0.097222 0.095394

copied Gosset G21 halfcube halfcube

1.000000 0.142857 0.072072 0.070812

induced radii and jump points used

I

0.540061

0.5

2 3 1 1 1

0-547722 0.500000 0.612372 0.918417 1.618033

3-3 3-2 3-5 3-6 3-6

6 6 5 5

2 2 1 1

0.559016 0.577350 0.784464 0.929876

4-2 4-3 4-4 4-5

copied Steiner3cube Gosset Gosset

7 6 6

2 1 1

0.577350 0.849836 0.938510

5-2 5-4 5-5

0.075000 0.053146 0.053113

Gosset Steiner3cube Steiner3cube

8 7 7

2 1 1

0-612372 6-2 1.007220 6-4 1.031395 6-4

0.612372 0.884651 0.945249

6-2 6-3 6-4

0.0

0.972446 0.986597

6-1 6-4 6-4

0.042638 0.034197

Steiner3cube Gosset

7 8

1 2

1.068357 0.707106

1.033499 1.224744

7-3 7-3

1.008479 1.414213

7-2 7-3

0.577350

3-4

0.618033

3-5

0.0

4-1

0.0

5-1

0.0

3-1

0.612372

5-3

f

0-

3

B ? i

8-2

o

? ?

7-3 7-1

L.A. Sz&kly. N.C. W o d

368

quoted in this section took an hour or so of computation time. (The time required for Table 3 was much greater.) After computing Sing(k) and Nonsing(k) for 3 c k s n, the program made a list of all the jump points required to establish the upper bounds obtained for rn‘:’(m) and for rnik)(r) for large r (i.e. the element (r, b) of Nonsing(k) for which r was greatest). The results for n = 8, with 6 = 0, are shown in Table 4. The column headed “jump” gives a name to each jump point (r, b), “sing” shows Y for singular and N for non-singular and “r2” gives r2 if the jump point is precise and “?” if not. The name of the configuration used to establish the jump point is given, “copied” appears in the “configuration” column if the jump point was copied from the previous dimension, and “dim” gives the dimension of the configuration. Given a configuration, number the different distances appearing in it in increasing order. The one expandedlcontracted to equal 1 is given as “unit”, and the radii induced Table 5. Jump points ( r , b ) used in the proof of lower bounds on m y ) ( r ) for n s 9 and large r. r

n 3

0.0

3 3 3 3 3

0.500000 0.547722 0.577350 0.612372 0.918417

4 4 4

0.0 0 .!A7722 0.559016 0.577350 0 .?a4464 0.329876

4 4 4 4

-

Sing

r2

copied gosset G2 1 Steiner3cube gosset gosset

6 6 7 6 6

2 2 2 1 1

1 .oooooo 0.142857 0.075000 0.053146 0.053113

copied Steiner3cube gosset Steiner3cube Steiner3cube

7 8 7 7

2 2 1 1

-

0.069444 0.042638 0.034197

n2cube Steiner3cube gosset

9 7 8

2 1 2

N

3

Y

ce

0.034604 0.028820

Steiner3cube n2cube

9 9

1 2

?

5

6 6 6 6 6 6

0.0 0.577350 O.blZ342 0.612372 0.884651 0.945249

N N

7

0.0

7

0.612372 0.707106 1.007326 1.033499

N N

ca

2 2 3 1 1 1

1.000000 0.222222 0.20?920 0.142857 0.072072 0.070812

N

3

0

1.295163

4 5

2 2 2 1

N

0

3/10 5/16 1/3

5/16 1/3 3/8

9

1

5 6 6 5 5

N Y Y

N

9

3

copied halfcube gosset C21 half cube halfcube

N

7

N Y

ce

2

1.000000 0.312500 0.222222 0.209773 0.097222 0.095394

N

Y

0.707106 1.079198

4

copied perm2 t 3 halfcube n2cube cross polytope cross polytope pentagonal

N

0.0

8 8

unit

1.000000 0.400000 0.312500 0.316216 0.142857 0.137931 0.128000

Y N

0.559016 0.577350 0.612372 0.849836 0.938510

8

dim

copied copied perm;! t 3 copied simplex morning star

0

1/4 3/10 1/3 3/8

5

7 7

configuration

1.000000 0.500000 0.400000 0.333333 0.250000 0.187500

N N

5 5 5 5

7

bound

Y

N

7

N

1

N Y

N N

Y

0 1/3 3/8 3/8 3

7

0

3/8 1/2

N N

7

Y

1/2

N Y

?

7

5 4 4 4

1

Boundr on the measureable chromatic number of R"

369

by the others are shown separately (preserving the order), together with the jump point used to estimate mik-')(r1)for each induced radius r,. Briefer results, omitting the induced radii, are given in Table 5 for n = 9, again with 6 = 0. In Table 6, we give for 6 = O.OOO1 the nonsingular jump points with greatest radii, for all dimensions up to 24, and also the singular jump points with r = to, where these give an improved bound (i.e. in all dimensions but 2, 3, 5 , 6 and 7). The implied lower bounds on %,,,(S(")(r))are also listed, as well as the configurations which gave these jump points. We do not give a proof as in Table 5 because thousands of jump points were involved. The configurations listed in Section 5 which were found by the computer to be useless in establishing these bounds were: Steiner-4-cube, Steiner 4-2 cube, "other" polytope, S(5, 8, 24), S(5, 6, 24), S(4, 5, l l ), perm(-1, 1,0,0,0), S(5, 4,23), S(4, 7, 23), Gosset 8 X Si (i = 1, 2, 3), Gosset 6 x Gosset i (i = 7, 8), Table 6. Our best upper bound b on rnp'(r) for large r. Its reciprocal gives a lower bound 1.b. on X,,,(S(")(r)). n

r

b

1.b.

2 3 4 4 5 6 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24

0.81941 0.91841

0.28571428 0.18750000 0.12800000 0.13793103 0.09539473 0.07081295 0.05311365 0.03419769 0.04264237 0.02882153 0.03460966 0.02234835 0.02850742 0.01789325 0.02404499 0.01437590 0.01902089 0.01203324 0.01490547 0.00981770 0.01301242 0.00841374 0.00990294 0.00677838 0.00847505 0.00577854 0.00767212 0.00518111 0.00680895 0.00380311 0.00515955 0.00318213 0.00436088 0.00267706 0.00329095 0.00190205 0.00228112 0.00132755 0,00160037 0.00107286 0.00129086

4 6 8 8

Do

0.92987 0.93851 0.94524 1.03349 ca

1.05654 ca

1.12915 ca

1.44913 ca

1.85404 ca

2.61007 ca

3.93276 Do

2.59807 Do

8.04155 Do

5.88784 Do

6.45497 Do

5.88784 on

5.88784 Do

7.22649 ca

6.03845 ca

8.59909 Do

8.34770 ca

10.9457

11 15 19 30 24 35 29 45 36 56 42 70 53 84 68 102 77 119 101 148 118 174 131 194 141 263 194 315 230 374 304 526 439 754 625 933 775

configuration

morning star pentagonal cross polytope halfcube Gosset Steiner3cube Gosset Steiner3cube n2cube Steiner3cube n2cube quartercube n2cube n2cube n2cube G6xG6 nlcube Steiner3,6/8cube n2cube G7xG7 n2cube Steiner3,6/8cube n2cube Steiner3,6/8cube n2cube L44 44443 n2cube L444444 n2cube Steiner3,6/8cube nlcube L3445 nlcube L333 L23 L35 L2 L2 linspace linspace

dim

3 4 4 5 6 7 8 7 9 9

10 10 11 11 12 12 13 13 14 14 15 15 16 15 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24

3m

L.A. SzCkely, N.C. Wormold

G22 and L, as well as those indicated in Tables 2 and 3. Not all of the remaining general configurations were useful in all dimensions. For, example, the n2-cube was used of dimensions 4, 5 and 9 to 21, but the n3-cube was only used of dimensions 10 and 11.

8. Open problems 1. Is miz)s !? The method of the present paper does not seem strong enough to solve this. 2. What is inf,,,,,mg)? This is closely related to the following question of Erdos: what is maxlH,,lX(GH) in the plane? It is known that these quantities can be about l / r G ( r G ) , but it is not known if they have polynomial bound. For example, the best known results for r = 2 are

(See Proposition 1.2(a), (b) and Corollary 2.5 in [34].) 3. Is it possible to use our method to prove an exponentially small upper bound for my)? If the answer is yes, the configurations involved must have strong design-like properties. Is an improvement of (1.2 + o(l))-” possibly this way? Is there an E > 0 and a configuration in BB“ of at least (1 B)” vertices, such that the shortest distance occurs at least in B% of the distances? If the answer is yes, mathematical induction may work. 4. What is the connection begween the structure of H and mg)? There are some results for n = 2. The first author proved [34] mg!b)=s if b / a 2 1.401 and conjectured [351 that

+

r n Z ) = O ifsupH=m,

(5.1)

Using a weaker concept of upper density (measured on concentric cubes around the origin) Weiss [37] proved (5.1) by ergodic theory. An alternative proof was given by Falconer and Marstrand, who also proved (5.2) (see [12, 10, 131). Furstenberg conjectured the generalization of Weiss’s theorem: “If a set of the plane has positive upper density measured in concentric cubes centred in the origin, then it contains vertices of a regular triangle with every long enough edge length. Further, it contains every big enlargement of every finite planar configuration”. The same conjecture will be formulated for the upper density concept used by us as well. (5.1) is a special case of the conjecture for a two point configuration.

Bow& on the measureable chromatic number of Iw"

371

5. How can one construct large 1-independent sets in R"? This seems to be related to the sphere-packing problem in R" (see [23], Assertion 2). On the other hand, even in R2, the densest sphere-packing does not give the densest 1-independent set (see Croft [8]). In R3 (from a table of Leech [25]), putting spheres of radius f at the centres of spheres in a best-known packing yields a set of density 10.80-'. Using the same table and a similar construction yields the following densities in higher dimensions n: n = 3 :10.80-' n = 4 :25.83-' n = 5 : 82.54-' n = 6: 171.62-' n = 7 :1634.lo-'. The growth here is faster than exponential. Another approach to proving rnP)al/k is to prove the existence of a measurable k-colouring of Gl in R". The bound rnP) 3 (3 o(1))-" can be obtained by this means, but the function o(l), coming from Butler [2], is hard to compute for small values of n. 6. What happens to xm and if the distances in R" are defined by another norm? 7. In the definition of measurable chromatic number we may replace the Lebesgue measurable sets by any other a-algebra and ask for the corresponding restricted chromatic number. The question looks the most meaningful for sets having the property of Baire (see Oxtoby [29]) because of the deep analogy between them and the Lebesgue measurable sets. Is it true that the chromatic number x b , in which the colour classes must have the property of Baire, is always equal to xm?

+

ME)

Acknowledgement The blocks of S(5,8,24) were obtained with the kind help of Marston Conder using Cayley.

References [l] N.G.de Bruijn and P. Erdiis, A colour problem for infinite graphs and a problem in theory of relations! Nederl. Mad. Wetensch. Proc. Ser. A. 54 (1951) 369-373. [2] G.J. Butler, Simultaneous packing and covering in Euclidean space, Proc. London Math. Soc. 25 (1972) 721-735. [3] P.J. Cameron and J.H. van Lint, Graphs, codes and designs, London Math. SOC. Lecture Notes Ser. 43 (Cambridge University Press, 1980). [4] J.H. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969) 79-88. [S] H.S.M.Coxeter, Regular Polytopes (Dover Publ. Inc., New York, 1973). [6] H.S.M. Coxeter, Regular Complex Polytopes (Cambridge University Press, London, 1974).

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171 H.S.M. Coxeter, The pure Archimedean polytopes in six and seven dimensions, Proc. Camb. Phil.SOC.24 (1927) 1-9. [S] H.T. Croft, Incidence incidents, Eureka (Cambridge) 30 (1%7) 22-26. [9] P. Delsarte, J.M. Geothals and J.J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977) 363-388. [lo] P. Erdiis, Some combinatorial, geometric and set theoretic problems in measure theory, in: Measure Theory, Oberwolfach 1983, Lecture Notes in Mathematics 1089 (Springer-Verlag. 1984) 321-327. 111) P. Erd6s and C.A. Rogers, Covering space with mnvex bodies, Acta Arithmetica 7 (1962) 281-285. 1121 K.J. Falconer and J.M. Marstrand, Plane sets with positive density at infinity contain all large distances, Bull. London Math. SOC.18 (1986) 471-474. [13] K.J. Falconer, The realization of small distances in plane sets of positive measure, Bull. London Math. Soc. 18 (1986) 475-477. [I41 K. J. Falconer, The realization of distances in measurable subsets covering W", J. Combinat. Theory A 31 (1981) 187-189. [ 151 P. Frank1 and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (4) (1981) 357-368. [I61 D. Gorenstein, Finite Simple Groups: An Introduction to Their Classifications (Plenum Press, New York-London, 1981). 1171 T. Gosset, On the regular and semi-regular figures in space of n dimensions, Messenger of Mathematics 29 (1900) 43-48. 1181 H. Hadwiger, Ein Uberdeckungssatz fur den Euklideschen Raum, Portugaliae Math. 4 (1944) 140-144. 1191 H.Hadwiger, Ungelhte Probleme No. 40, Elemente der Math. 16 (1%1) 103-104. [20] M. Hall, Jr.. Combinatorial Theory (Blaisdell Co., Mass., 1%7). [21] S. Helgason, Groups and Geometric Analysis (Integral geometry, Invariant differential operators and Spherical functions) (Academic Press, 1984). [22] Kai Lai Chung, On the probability of the Occurrence of at least rn events among n arbitrary events, Ann. Math. Stat. 12 (1941) 328-338. [23) D.G. Larman and C.A. Rogers, The realization of distances within sets in Euclidean space, Mathematika 19 (1972) 1-24. [24]J. Leech, Some sphere packings in higher space, Canadian J. Math. 16 (1964) 657-682. [25j J. Leech, Notes on sphere packings, Canadian J . Math. 19 (1%7) 251-267. [26] L. bv&, Combinatorial Problems and Exercises (North-Holland, Amsterdam-New YorkOxford, 1979). (271 L. Moser and W. Moser, Solution to Problem 10, Canad. Math. Bull. 4 (1%1) 187-189. I281 W.Moser, Research problems in discrete geometry (mimeographed) (1981). [29] J.C. Oxtoby, Measure and Category; A Survey of the Analogies Between Topological and Measure Spaces (Springer-Verlag, New York, 1971). [N]D.E. Raiskii, The realisation of all distances in a decomposition of the space R" into n + 1 parts (Russian), Mat. Zametki 7 (1970) 319-323. 1311 D.E. Rasiskii, the realisition of all distances in a decomposition of the space R" into n + 1 parts, Math. Notes 7 (1970) 194-1%. [32] A. Renyi, Foundations of Probability (Holden-Day, San Francisco-Cambridge-LondonAmsterdam, 1970). [33] L.A. Szekely, Inclusion-exclusion formulae without higher terms, Ars Combinatoria 23B (1987) 7-20. [34] L.A. Szdkely, Measurable chromatic number of geometric graphs and sets without some distances in Euclidean space, Combinatorica 4 (1984) 213-218. [35] L.A. SzCkely, Remarks on the chromatic number of geometric graphs, Graphs and Other Topics, Teubner-Texte zur Mathematik, Band 59 (Leipzig, 1983) 312-315. [36] P. du Val, On the directrices of a set of points in a plane, Roc. London Math. Soc. 35 (1933) 23-74. 1371 B. Weiss, Personal communication. I381 D.R. Woodall, Distances realized by sets covering the plane. J. Combinat. Theory A 14 (1973) 187-20.

Discrete Mathematics 75 (1989) 373-379 North-Holland

373

A SIMPLE LINEAR EXPECTED TIME ALGORITHM FOR FINDING A HAMILTON PATH Andrew THOMASON Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 ISD, England We give a simple algorithm which either finds a hamilton path between two specified vertices of a graph G of order n, or shows that no such path exists. If the probability of an edge in G is p L 12n-f the algorithm runs in expected time cnp-’ and uses storage cn, where c is an absolute constant.

It is well known that the problem of finding a hamilton cycle in a graph is NP-complete, and there is some value in having algorithms which are fast for most input graphs. Let %(n,p) be the probability space of graphs with n labelled vertices, where edges appear with probability p. Johnson [9] asked whether there exists an algorithm with linear expected running time, when the input is a randomly chosen element of % ( n , p ) . (The algorithm is allowed to enquire of an oracle which tells whether two specified vertices are adjacent; this commonly employed oracle models many situations, in which such an enquiry is cheap.) Gurevich and Shelah [7] answered the question by displaying an algorithm which runs in expected time cnlp, c being an absolute constant, and requires O(n’) storage, provided p is constant. In fact some of their argument works if p >> n-4. In this note we give another such algorithm, which has the advantages of being simpler, requiring only O(n) storage and working for p 3 12n-4. The order n / p for the expected time is best possible, as observed in [7],since in order to find the n edges of the cycle we expect to have to ask at least n / p questions of the oracle. Angluin and Valiant [l] gave an algorithm which almost surely finds a hamilton cycle in polynomial time if p 2 c(log n ) / n , and an improvement appears in Shamir [ll]. Finally BollobPs, Fenner and Frieze [3] gave an algorithm which is about best possible. One of the best known algorithms for finding a hamilton path [8] requires time n22”, so the problem is one of finding a fast algorithm which works on all graphs in % ( n , p ) except for a proportion somewhat smaller than 2-“. This will be our algorithm A2. A very fast algorithm, which works on most graphs but not on as many as A2 does, is used to bring the running time down to O ( n / p ) . This will be algorithm Al. The algorithm A2 was originally based on a study of hamilton cycles in “concrete random graphs” in [12]. 0012-365X/89/$3.500 1989, Elsevier Science Publishers B.V. (North-Holland)

374

A. Thomason

We shall suppose that the algorithm has to find a hamilton path in an input graph G between two specified vertices s and t. (An easy modification gives the hamilton cycle version.) The graph G will be randomly chosen from %(n, p), with p 3 12n-i and n large. If Q is an s - I path and u and u are vertices on Q , we write UQUto denote the section of Q between u and u . We say u precedes u on Q if when traversing Q from s to t the vertex u is encountered before the vertex v. We say an algorithm is successfur if it finds a hamilton s - c path or proves that G does not have one. It fuih if it gives up without succeeding. Our algorithm is as follows.

Step 1. Apply algorithm A l . Step 2. If algorithm A1 fails, apply algorithm A2. Step 3. If algorithm A2 fails, apply algorithm A3.

If at any stage algorithm A1 has made more than 13n/p probes on edges, it fails. Step 1. Set Q = s. Step 2. Repeat whilst n - lQl k 14nil: replace the path Q = sQw by sQwx where x is a neighbour of w in G - Q - t. If no such neighbour is found, algorithm A1 fails. Step 3. Let Q =sQv. Find a set Y of 13.41 neighbours of u in G - Q - t and a set Z of [3n'] neighbours of t in G - Q - Y . Find an edge y z , y E Y , z E 2. Replace Q by sQuyzr. If the sets Y, 2 or the vertices y, z cannot be found, algorithm A1 fails. Step 4. Repeat until Q is a hamilton s - t path: Choose w E G - Q . Find two neighbours u and u of w which are adjacent on Q. If no such pair exists, algorithm A1 fails. Replace Q by sQuwuQt. In order to present the main algorithm we need to define a s2-mutching in a bipartite graph with vertex classes X and Y.This is a set of edges F such that each vertex of X is incident with at most two edges of F and each vertex of Y is incident with at most one edge of F. A maximum SZmatching is a ~Zmatching with the maximum possible number of edges. Algorithm A2

Step 1. Find V,, the set of all vertices in G of degree at most pn/4. If IV,l 3 24p-' algorithm A2 fails. If V, = B go to step 5.

Simple linear expected time algorithm

375

Step 2. Construct a maximum s2-matching from V, to G - V,- {s, t } . Let V, be a set of 2 IV,l vertices of G - V, - {s, t } containing the vertices of degree one in this s2-matching. Step 3. Let H be the subgraph of G induced by V, U V, U {s, t } . Form H* from H by joining every pair of vertices in V ( H * )- V,. Step 4. Find a hamilton s - t path Q* in H*, using any algorithm such as A3. If no such path exists, then G has no hamilton s - t path and algorithm A2 succeeds. Let k = [ 2 m 1 . Given vertices u, v in G - V, and a subset X c V ( G ) , a 3-path in G - X is a path uulvlu,found by choosing k neighbours of u in G - X - {v}, a disjoint set of k neighbours of v in G - X - { u } , and finding some edge between these sets. Step 5. If V, # 8, construct from Q* an s - t path Q in G with V, c V ( Q ) as follows: if uQ*v is a maximal subpath of Q* contained in E ( Q * ) E ( G ) , replace uQ*v by a 3-path uulvlvin G - V ( Q * ) , and call the new path Q * . Repeat whilst E ( Q * ) - E ( G ) Z0, then put Q = Q * . If V, = 0, let Q be a 3-path ssltlt in G . If at some stage a required 3-path is not found, algorithm A2 fails. Step 6. Repeat whilst lQl S 3k: select adjacent vertices uv on Q with u, v E G - V,, and replace uv in Q be a 3-path in G - V ( Q ) . If no such 3-path is found, algorithm A2 fails. Step 7. Repeat until Q is hamilton s - t path: choose w E G - Q . Construct 2k vertex disjoint paths Pl, . . . ,Pu, of length at most two from w to Q - s as follows. Choose 5k neighbours of w . If 2k of these lie on Q - s, these give P,, . . . , P2k. Otherwise select 3k neighbours of w not on Q , and find a maximal independent set of edges between these neighbours and Q - s. These give P,, . . . ,Pu, unless there are less than 2k independent edges, in which case algorithm A2 fails. Let { x l , . . . ,X 2 k ) be the end vertices of these paths and let { y , , . . . ,y2k} be their predecessors on Q . If there is an edge yiyj, replace Q by sQyiyjQxiewP+jQt. If there is no edge yiyj, algorithm A2 fails.

Algorithm A3 Any algorithm which always succeeds in time 2"+"(") and in space O ( n ) may be used here. It is shown, e.g. in [7],that the dynamic programming algorithm of [8] and the proof of Savitch's theorem [lo] can be blended to this end. Thorem. Algorithm A is successful. The expected running time is at most cnlp, provided p 3 12n-i, and the storage is at most cn, where c is an absolute constant. Proof. The proof follows directly from Lemmas 1 and 2 below.

A. Thomason

376

Lemma 1. The algorithm A 1 succeedr with probability at least 1 - 2-75'ppn-2. The time used and storage required are bounded by cnlp and cn respectively, where c is an absolute constant.

Lemma 2. The algorithm A2 succeeds with probability at least 1 - 2-2n+o(n).The time used and storage required are bounded by ~ ( 2 ' ~+ 'n'/p) ~ and cn respectively, where c is an absolute constant. Before proving these lemmas, we define P(n, p , k) to be the probability of scoring at most k successes in a sequence of n independent Bernoulli trials, each with probability p of success. If k = ( 1 - p ) p n with p > 0 then

P(n, p , k) < e-p2pnn

(-F)

(see f l ] ;this follows from ChernoB's bound [4],or see [2, chapter 11).

Proof of Lemma 1. A graph G in %(n, p ) can be thought of as being generated by the following process. For each pair u,u E V ( G ) , conduct b independent Bernoulli trials, each with probability p ' of success, where ( 1 -p')* = 1 - p . If any trial is successful, the edge u v is inserted; if all are unsuccessful, the edge uv is let out. Now algorithm A 1 never attempts to establish the non-existence of a hamilton path, so the probability of its failure on a random input graph is simply the probability that it fails to find a hamilton path. Since the algorithm questions the oracle at most three times about any edge, an upper bound for the probability of failure of the algorithm can be obtained under the assumption that the answers to each question are independent of previous answers, and each answer is "yes" with probability p' > p / 3 , where ( 1 - P ' ) ~= 1 - p. The constraints on time and storage are manifest, so we just check the first claim of the lemma. Let I = 14nfI - 1. The probability that some vertex w in step 2 fails to have a neighbour x is at most (1 -p')'. Hence the probability of step 2 failing to work after 4np-I probes is at most

p(4np-',p', n

-

[4n51)+n(l -p')'<e-"/24+ne-p"3
by (t), where p o = 2-75'fpn-2/3. In step 3 , the probability that Y or 2 does not exist, or that there is no Y - Z edge, is at most

2P( 14n;l - 13n;] - 1, p ' , 13n4J)+ ( 1 -P')~"' < 3e-5nf
+

Simple linear expected time algorithm

377

Thus the probability that algorithm A1 fails, or needs 13np-' probes, is at most 3p0. 0 Algorithm A2 is based on the fact that, with a very high probability, the graph G is close to being (p, 2l/&)-jumbled. A graph G is said to be ( p , a)-jumbled, where O < p < 1s a, if for every induced subgraph H of G , (e(H)- p ( l ~ l ) ls a (HI holds. This definition was introduced in [12], where it is shown that ( p , a)-jumbled graphs behave in many ways like random graphs. Easily checked conditions implying a graph is ( p , @)-jumbled are given in [12]. A theorem of Chvgtal and Erdiis [5] is used in [12] to find hamilton cycles in ( p , a)-jumbled graphs and the same ideas form the basis of the algorithm here. In fact we do not need the full strength of the definition of a ( p , a)-jumbled graph, and prove only the following lemma. Lemma 3. Let k = [ 2 m 1 . The probability that a graph G E %(n, p ) contains two disjoint sets of vertices Y and Z , JYI = IZI = k, with no edge between Y and Z , is at most 2-2n+o(n).In particular G is unlikely to have an independent set of 2k vertices. Proof. The probability that such sets Y and Z exist is at most n2(l - P ) ~ * < n 2 k e - ~ k 2 < 2--2n+o(n). 0 Proof of Lemma 2. The times required for steps 1 and 3-7 are clearly bounded by functions of orders c1n2,c3n2,~ 4 2 ~(since ~ " IH*I S 3 lVol + 2), c5 IVol n2, c,nk2 and c7nk2 respectively, and the storage by cn for each step. Moreover to find the ~Zmatchingin step 2, construct a bipartite graph B from V ( G )- V, - { s , t } and two copies of V, by joining each vertex in each copy of V, to the vertices of V ( G )- V, - {s, t } which are its neighbours in G . There is a 1-1 correspondence between matchings in the bipartite graph B and SZmatchings in G from V, to G - V, - {s, t } . Using say the matching algorithm of Edmonds [6], a maximum matching in B can be found in time order IVol (IV,l+ IE(B)I) and store order IV,l + IE(B)I. Hence step 2 can be completed in time c2n/p and with storage cn. So only the first assertion of the lemma now requires proof. In [12] it is shown that if G is ( p , 2G)-jumbled then IV,l < 1 4 m . However, we require V, to be somewhat smaller so we give a separate argument. The graph G can be thought of as being generated by selecting neighbours for each vertex with probability p ' , where (1 - P ' ) ~= 1- p , and identifying double edges. The probability that a given vertex has degree less than p ' n / 2 is by (t) at most e-P'(n-1)'8. Thus the probability that IV,l L24p-l is at most n24/pe-24p'(n-1)'8p < e-7d5 if n is large, so step 1 fails with probability less than 2-2n+o(n) Steps 2 and 3 require no comment, and to verify step 4 we need only show that if Q* doesn't exist then G has no hamilton s - t path. So suppose G does have a

378

A. Thomason

hamilton s - t path. Let G* be the graph obtained by joining every pair of vertices in G - V,. Then G* has a hamilton s - t path Q * * . Now E(Q**)\E(G V,) consists of a set of disjoint paths in G - E(G - V,), containing between them all of V,, whose endvertices lie in (C - V,) U {s, t } . Call the set of edges spanned by such a set of paths a kernel set. Choose a kernel set containing as many edges as possible from the ~Zmatching.The kernel set splits V, as W, U W, U W,, where is the subset of V, joined by j edges of the kernel set to G - V,. Let K be the set of vertices of G - V, joined by the kernel set to V,. They any vertex in y.matches to at most 2 -1 vertices in V,\K, for otherwise if x E W, and xy is in the S:Zmatching, where y $ K . we can remove a kernel set edge from x, replace it with xy, and create a kernel set with more edges from the SZmatching. Now, for each vertex in K, choose an edge of the kernel set incident to it. Then a vertex of W, is incident with at most j of these edges. So these edges, along with the edges of the SZmatching meeting V,\K, together form a SZmatching of order IV, U K1. Since the largest SZmatching has order V,, we see that K IV,i E V,. It now follows from the definition of a kernel set that we can construct a hamilton s - t path Q* in H*, as claimed. To verify step 5, note u and v have degree at least p n / 4 in G . So if X is any set of vertices with 1x1 6 6 I&( we can find sets Y of neighbours of u and 2 of neighbours of v in G - X - { u , u } , with IYl, l Z l 2 p n / 4 - 1 - 6 1 V 0 I . Since p n / 4 - 1 - 6 I V,l 3 2k, the desired disjoint set of k neighbours for each of u and v can be found, after which Lemma 3 implies the existence of the 3-path we seek. lQl S 3k. Note that u , v can For step 6 we just repeat this procedure with be found since IV,l > IVol + 1. All that remains is to check that Step 7 succeeds with high probability. Now if no 2k of the 5k chosen neighbours of w lie on Q - s, let A be a set of 3k neighbours not on Q. Let X be the set of vertices covered by the maximal set of independent edges from A to Q - s. Putting W = A - X , Z = V ( Q )- s - X and noting there is no W - 2 edge we see by Lemma 3 that IW 1 < k so IA fl XI 3 2k. These 2k edges give us the paths PI, . . . , P2k. The existence of some edge y,y, is again implied by Lemma 3. This completes the proof of Lemma 2. I7

1x1

Remrulrs. It is simple to mod@ the algorithm to find a hamilton cycle. In algorithm A1 we change step 3 to find a path vyzs. In algorithm A2 we find at step 4 a hamilton cycle Q*. The rest is straightforward. In fact algorithm A2 can be used to find Bpn edge disjoint hamilton cycles, for some B > O , if a modification to Lemma 3 is allowed forbidding large subgraphs of low degree. What happens if G is a directed graph? It is clear that algorithm A1 can be trivially modified, and in fact given a suitable definition of a GZmatching (at most one edge in each direction at EX) the only problem with algorithm A2 comes in Step 7. In this case we find many paths in each direction between w and Q, and so find subsets A, B, C and D of V ( Q ) , each of order such that A

m,

Simple linear expected time algorithm

379

precedes B, B precedes C, C precedes D and such that either (i) w sends a path to each vertex in A and receives one from each vertex in C or (ii) w receives a path from each vertex in A and sends one to each vertex in C. We may as well suppose B and D are paths. We then prove a lemma which tells us that almost surely, if X, Y, Z are sets of order and Y is a path, then there is an edge uv in Y and edges ux and zv, x EX,z E Z. Applying this lemma to C+,B, A - in case (i) or to A+, D, C - in case (ii) (where A-, A+ are the predecessors, successors of A) allows us to extend Q to incorporate w. These operations all work provided we increase the lower bound of 12 for p n f to some larger constant. However, it now requires k3 rather than k2 probes actually to find the modification to Q, and so the time required for algorithm A2 increases to cng-3. Nonetheless the main theorem still holds.

References [l] D. Angluin and L.G. Valiant, Fast probabilistic algorithms for hamilton circuits and matchings, J. Computer and System Sci. 18 (1979) 155-193. [2] B. BollobBs, Random Graphs (Academic Press, 1985). [3] B. BollobBs, T. Fenner and A. Frieze, An algorithm for finding Hamiltonian cycles in random graphs, Combinatorica 7 (1987) 327-341. [4] H. Chernoff, A measure of asymptotic efficiencyfor tests of a hypothesis based on the sum of observations, Ann. Math. Stat. 23 (1952) 493-509. [5] V. ChvBtal and P. Erdbs, A note on hamiltonian circuits, Discrete Math. 2 (1972) 111-113. (61 J. Edmonds, Paths, trees and flowers, Canad. J. Math. 17 (1965) 449-467. [7] Y. Gurevich and S. Shelah, Expected computation time for hamiltonian path problem and clique problem, SIAM J. Comp. 16 (1987) 486-502. [8] M. Held and R.M. Karp, A dynamic programming approach to sequencing problems, J. Soc. Indust. Appl. Math. 10 (1%2) 196-210. [9] D. Johnson, The NP-completeness column: an ongoing guide, J. Algorithms 5 (1984) 284-299. [lo] W.J. Savitch, Relationships between nondeterministic and deterministic tape complexities, J. Comput. System Sci. 4 (1970) 177-192. [ l l ] E. Shamir, How many random edges make a graph Hamiltonian? Combinatorica 3 (1983) 123- 132. [12] A. Thomason, Psuedo-random graphs, Annals of Discrete Math. 33 (1987) 307-331.

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Discrete Mathematics 75 (1989) 381-386 North-Holland

381

DENSE EXPANDERS AND PSEUDO-RANDOM BIPARTITE GRAPHS Andrew THOMASON Dept. of Pure Muthematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 lSD, U.K.

1. Introduction There has been considerable interest over the last few years in bipartite graphs wherein any set of vertices is guaranteed to have quite a large set of neighbours, without the maximum degree being large. Formally, it is customary to define a bipartite graph, with vertex classes X and Y , to be (n,a, 6)-expanding if = IYI = n and every subset A c X with IAJ= a has at least b neighbours in Y; b is usually a function of a. The aim is to find graphs with b as large as possible, given some constraint on the maximum degree. It is not hard to show that random bipartite graphs will have more or less best possible expansion properties, but explicit constructions have proved hard to find. Most attention has been focussed on linear expanders, that is, where the maximum degree is regarded as constant whilst n grows large (see for instance Margulis [lo], Chung [6] and Gabber and Galil [7]). However, dense expanders, where the maximum degree grows with n, have also found applications, most noticeably to parallel sorting algorithms. The algorithm of Ajtai, Koml6s and SzemerCdi [l] for sorting n objects in time O(1ogn) using n / 2 processors uses expanders explicitly, and the proofs of the two round sorting algorthms of Haggkvist and Hell [a] and of Bollobis and Thomason [5] make implicit use of the expanding properties of certain dense graphs, in the latter case random graphs. (An algorithm for sorting n objects in r rounds, using rn parallel processors, makes rn pairwise comparisons in the first round, deduces as much as possible from the result via transitivity, then makes rn further comparisons in the second round, and so on. The rth and final round consists of comparing all pairs whose relative order remains in doubt. We denote by $(n) the least value of rn for which there is such an algorithm. Moreover $(n, d) denotes the least rn for which there is an algorithm employing only d-step transitivity between rounds; this means if we know x,, <x1 < - * < x k we may deduce xo < x k only if k s d . ) It was shown in [5] that

1x1

-

(2/G+ o(l))n$s f i ( n ) d (1/2)& log n and 2-7n l+dl(2d- 1) s f i ( n ,

d ) d (1/2d)nl+d’(W--1)(log n)l’(U-’)

0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V.(North-Holland)

382

A. Thomclson

for b e d d and large n. The proofs were non-constructive. Alon [2], developing an eigenvalue technique of Tanner [ll],was able to prove that a certain graph (which we describe later) was (n, x , v - nl+”d/x)-expandingfor all x , and showed how, in the case d = 4, such a graph would yield a constructive proof that f(n, 2) = O(n3). For a good survey of the use of graphs in parallel sorting, see Bollobiis and Hell [4]. Our p u p e in this note is to point out how checking a very simple condition often suffices to show that a dense bipartite graph is a good expander. The check is much easier to apply than the eigenvalue method, though in the cases where both methods are feasible both will give much the same results. The sufficient condition we offer is derived from a study of pseudo-random graphs [12]; for a survey see Thomason [13] and for an extension to hypergraphs see Haviland and Thomason [9]. The check involves merely the degrees of vertices and the number of common neighbours of pairs of vertices. In fact it is sufficient to imply the bipartite graph is “pseudo-random” in a sense analogous to that of [12] and [9], but we shall not develop the point.

2. SIldlirient conditioas

Our principal result is Theorem 2. The proof is very straightforward and is the bipartite analogue of Theorem 1. The latter theorem appeared in [12] but the proof was slightly deficient, so we give a correct version here. Theorem 1. Let G be a graph of order n, with minimum degree at least pn, where 0 < p < 1. Let p 3 0 be such that no two vertices of G have more than p2n p common neighbours. Then, for every induced subgraph H of G ,

+

hoIa3. Here 2a = E + v p m ’ , and

E

= 1 ifp IHI

< 1, E = 0 otherwise.

Proof. Let H be a subgraph of G of order k s n, and let the average degree in H be d. Let a,, . . . , ak be the degree sequence of H, and let bl, . . . ,bn-k be the number of edges between H and each of the n - k vertices of G - H. Then

ai = kd and I=

k

n-k

k 1

bi 2 i=1

(pn - ai) = k(pn i= I

- d).

Moreover, since no two vertices have more than p2n + p common neighbours, we have

Dense expanders and bipartite graphs

383

so, if p n 2 d,

Rearranging gives

(d - p ( k - 1))*d [(n- k ) / n ] [ ( k- 1 ) p + np(1 - p ) ]

c k p + (n - k ) p ( l - p ) spn+pk

+ 2p(d - p k )

+ 2p(d - p k )

+p2

+p2

ifpnzd,

which yields the result claimed. We must now check the result in the case p n c d. Since bi 5 0 the above gives

k(

); (i ) ( p 2 n+ p ) 6

or d s d m ( p 2 n + p ) + $ + f,

(1)

where m = k - 1. Our aim is to show Id - p m l < 2a, and since d a p n , we must show d s p m + 2 a . So by ( 1 ) it is enough to demonstrate v m ( p 2 n + p ) + $ s p m 2a - f; squaring both sides (note p m + E 2 d) this is equivalent to proving

+

p ( n - m ) ( p m - 1) - p s (2pm + 2~ - 1 ) V p m+ 2 ~ p m . This is immediate if p m C 1, the left hand side being negative. Otherwise E = 0 and 2pm - 1 z p m , so it suffices to show p ( n - m ) C d p z . In this case p n z p m 3 1, and we compute from ( 1 ) and p n s d that pm >pn(p(n - m ) - 1). So we need only verify p ( n - m ) C p d m , and this is clearly true. 0 For the bipartite analogue of this theorem it is convenient to use the notation e ( A , B ) to denote the number of edges between the vertex subsets A and B.

Theorem 2. Let G be a bipartite graph with vertex classes X and Y, where 1x1= IYI = n. Let each vertex iq X have degree at least pn, where 0 < p < 1, and let p 2 0 be such that no two vertices of X have more than p2n + p common nejghbours. Then f o r every subset A c X and every subset B c Y,with /A]= a and 14= b, le(A, B ) -pub1

S

where E = 1 i f p a < 1 and

E

Eb + d a b ( p n + pz), = 0 otherwise.

Proof. The proof is very similar to that of the previous theorem. Define d by e(A, B ) = bd. Then e ( A , Y - B ) 5 apn - e ( A , B ) = b(rpn - d ) , where r = a/b. By estimating common neighbours of pairs of vertices of A, we have, if rpn 2 d ,

A. Thomason

384

Rearranging gives

(d -pa)’S [r(n- b)ln][(a- 1)p + np(1- p ) ] r(pn + P I , as desired. Otherwise d 2 rpn, and we obtain instead b(

i)

C

( i ) ( p h + p ) or d d d r m ( p 2 n+ p ) + 5 + 4,

(2)

+ v-.

where M = a - 1. Our aim is to show Id -pal s /3, where = E Since d 2 rpn S p a this means we show d s p a j3. In fact by (2) it is enough to demonstrate
+

p(rn - m ) ( p m - 1) - r p G (2pm

+ 2.5 - 1)d-

+

+2 ~ p m .

The right hand side is positive so we need only consider the case p m 2 1. But then E = 0 and 2pm - 1 a p m , so it suffices to show p(rn - m ) s d z . Moreover p m 2 1 implies rpn 2 p m 1, whence ( 2 ) and rpn s d yield rpm t rpn(rpn - p m - 1). So it need only be shown that p(rn - m ) s p V r n ( r n - m), which is manifest. We remark that in each of the proofs approximations were made for the sake of obtaining a cleanly stated inequality, designed for ease of general application. In any particular circumstance it would be possible to obtain a better, though likely not significantly better, result. The expansion properties of the graph described in the last theorem can be stated more explicitly.

corollary 3. Let G be a bipartite graph with vertex classes X and Y , where 1x1 = I Yl = n. Let each vertex in X have degree at least pn, where 0 < p < 1, and let p 3 0 such that no two vertices of X have more than p2n + p common neighbours. Then G is ( n , a, n - nlpa

- plp2)-expanding for every a S n.

Proof. If pa s 1 there is nothing to prove. Otherwise Theorem 2 shows G is (n, a, n - b)-expanding, where pab S d u b ( p n + pa), as claimed. 0 3. Some examples Here are just a few examples of how Corollary 3 may be used to check the expansion properties of a graph. (1) Let G be a random bipartite graph, with edge probability n-6 where S < $. Standard estimates for the binomial distribution show that G satisfies the conditions of the corollary with p = n-’(l+ o(1))and p = n*-’(l+ o(1)). So G is

385

Dense expanders and bipartite graphs

(n, a, n - [n’+’/a + n*+’](l + o(1)))-expanding. Of course, in this case it would be better to use the usual methods of random graph theory. (2) Take a projective geometry of dimension d over the finite field of order q. Let X be the points and Y the hyperplanes of the geometry. Form G by joining x E X to y E Y if x ~ y Then . it is straightforward to compute n = (qd+’ - 1)/ (q - l), that G is (qd - l)/(q - 1)-regular, and every two points lie in (qd-l - 1)/ (q - 1) hyperplanes. Calculation then reveals that G satisfies the condition of the corollary with p = n-*(1+ o(1)) and p = 0, so G is (n, a, n - (1 + o(l))n””d/a)expanding. The o(1) term can be removed by following through the proof of Theorem 2 for this particular graph and being less wasteful. This is the graph mentioned earlier, used by Alon [2]. (3) Take a symmetric block design with parameters v, k, A. Let X be the set of v objects and Y the set of v blocks, and join x E X to y E Y if x E y. This gives us a pn-regular graph, where n = v and pn = k. Since each pair of objects occurs together in A blocks, we have A = k(k - l)/(v - 1)
+

References

[l] M. Ajtai, J. Koml6s and E. Szemeredi, Sorting in C logn parallel steps, Combinatorica 3 (1983) 1-19. [2] N. Alon, Eigenvalues, geometric expanders, sorting in rounds and ramsey theory, Cornbinatorica 6 (1986) 207-219. [3) E. Artin, Geometric Algebra (Interscience Publishers, New York, 1957). [4] B. Bollobis and P. Hell, Sorting and graphs, in Graphs and Order, I. Rival, ed., NATO AS1 series (Reidel, 1985) 169-184. [5] B. Bollobh and A. Thomason, Parallel Sorting, Discrete Appl. Math. 6 (1983) 1-11. 161 F.R.K. Chung, On concentrators, superconcentrators, generalizers and non-blocking networks, Bell Syst. Tech. J. 58 (1978) 1765-1777. [7] 0. Gabber and Z. Galil, Explicit constructions of linear superconcentrators, J. Comp. and Sys. Sci. 22 (1981) 407-420. [8] R. Haggkvist and P. Hell, Parallel sorting with constant time for comparisons, SIAM J. Comput. 10 (1981) 465-472. [9] J. Haviland and A. Thomason, Pseudo-random hypergraphs, Discrete Math. 75 (1989) 255-278. [lo] G.A. Margulis, Explicit constructions of concentrators, Problerny Peredachi Informatsii 9(4) (1973) 71-80 (in Russian). English translation in Problems Info. Transmission (Plenum Press, 1975) 325-332.

386

A. Tlromapon

f l l ) R.M. Tanner, Explicit construction of concentrators from generalized N-gons, HAM J . Alg. Discr. Meth. 5 (1985) 287-293. [I21 A. Thomason, Pseudo-random graphs, in Proceedings of Random Graphs, PoznarE 1985, M. Karonski, ed.,Annals of Discrete Math. 33 (North-Holland, 1987) 307-331. [13] A. Thomason, Random graphs, strongly regular graphs and pseudo-random graphs, Surveys in Combinatorics, 1987, C. Whitehead, ed.. LMS Lecture Note Series 123 (1987) 173-195.

Discrete Mathematics 75 (1989) 387-404 North-Holland

387

FORBIDDEN GRAPHS FOR DEGREE AND NEIGHBOURHOOD CONDITIONS D.R. WOODALL Department of Mathematics, University of Noningham, Noningham NG7 2RD, England

For various graph-theoretic properties P that impose upper bounds on the minimum degree or the size of a neighbourhood set, we characterize the class %?(P‘) (%?(P‘)) of graphs G such that G and all its subgraphs (subcontractions) have property P. For example, if P is “ 6 Cxn” (6 = minimum degree, n =number of vertices, 0 < x < 1) then %?(P‘) = F(Kr+l), the class of graphs that do not have K,,, as a subgraph, where r = [1/(1- x ) ] .

1. Introduction In this paper we consider only simple graphs. When we contract an edge of a simple graph, it is understood that we remove one edge of each pair of parallel edges so formed, so that the new graph is again simple. A contraction of a graph is any graph obtained from it by successively contracting some of its edges, and a subcontraction (induced subcontraction) is a subgraph (induced subgraph) of a contraction, or, equivalently, a contraction of a subgraph (induced subgraph). We write H E G ( H G G ) if H is a subgraph (subcontraction) of G . If P is a property that a (simple) graph G may or may not have, let P‘ (pronounced “ P subgraph’ or “PSG’) denote the property “ G and all its subgraphs have property P”. Replacing “subgraphs” by, respectively, “induced subgraphs”, “contractions”, “subcontractions” and “induced subcontractions”, we obtain the properties Pa (“P induced subgraph” or “PISG’), P‘ (“P contract” or “PC”), P< (“P subcontract” or “PSC”) and PQ (“P induced subcontract” or “PISC”). Y?(P)(pronounced “the class of F’”) denotes the class of graphs possessing property P. If P “ @ Q e t (that is, Y?(P‘) = Y?(Q“)), then the properties P and Q are called subgraph-equivalent or SG-equivalent. In a similar way we define inducedsubgraph-equivalent or ISG-equivalent, contraction-equivalent or C-equivalent, subcontraction-equivalent or SC-equivalent, and induced-subcontractionequivalent or ISC-equivalent.

Example 1.1. Consider the following properties. P,: G is planar. P2:G # K5or K3,3. P3: G is not homeomorphic to K5or K3,3. 0012-365X/89/$3.500 1989,Elsevier Science Publishers B.V.(North-Holland)

D.R. Woodall

388

PT, and the well-known theorems of KuraThen P, UP: UP;‘ PT tjP; towski and Wagner can be interpreted as saying, respectively, that properties PI and P3are subgraph-equivalent , and PI and P2 are subcontraction-equivalent. Any characterization in terms of forbidden induced subgraphs can be expressed in this terminology: for example, Beineke’s characterization [I] of line graphs.

Example 1.2. Consider the following properties. PI: G is a line graph. P2: G is not equal to any of the nine graphs on page 131 of 111.

5: G # K1.3. Since P, ~3PF, Beineke’s theorem that P I G PF can be interpreted as saying that is one of Beineke’s nine properties Pl and P2 are ISG-equivalent. Since graphs and is a subgraph of all of them, it follows that Pl and P3 are subgraph-equivalent (hence, subcontraction-equivalent). But since the class of line graphs is not closed under subgraphs, this result does not characterize line graphs. Example 1.3. Consider the following properties, in which x(C), o(G)and a ( G ) denote the chromatic number, clique number and independence number of G. Pi: x ( G ) = o(G). Pz: cY(G)o(G)5 IV(C)l. P3: G # {G,c7, G, c9, G, * * .I. P4; G $ {CS, c7, c9, . .>. Ps: G # Cs. 1

The property of being perfect is P;i. LovAsz [2] proved that PI and P2 are ISG-equivalent. Berge’s Strong Perfect-Graph Conjecture is that P, and P3 are ISG-equivalent. It would follow from this, but is also an easy consequence of a theorem of Meyniel [4], that PI and P4 are subgraph-equivalent and (hence) Pl and Ps are subcontraction-equivalent. None of these results characterize property P,, but the first theorem and the conjecture (if true) characterize perfect graphs.

Example 1.4. Consider the following properties. P l : G is r-colourable. Pz: G has an independent set consisting of at least l l r of its vertices. P3: G # K , + , . Hadwiger’s conjecture can be interpreted as saying that, for each r 3 1, Pl and P3 are subcontraction-equivalent, which would imply that P I , P2 and P3 are all subcontraction-equivalent. This example is discussed in more detail in [7]. We shall write %=(GI,. . . , G,) for q P ‘ ) when P is the property G #G1, G 2 , .. . ,or G,,

Forbidden graphs

389

and analogously for %(P<), etc. In graph-theoretic properties, n, m and 6 will always denote the numbers of vertices and edges and the minimum degree of any graph to which the property applies. The purpose of this note is to characterize the classes %(P‘) and %(F) for various properties P that restrict the number of edges in a graph by imposing upper bounds on its minimum degree or the size of a neighbourhood set. In Sections 2 and 3 we consider characterizations of the classes F(K,+J and G<(K,+l)respectively, the latter class being the one that occurs in Hadwiger’s conjecture. In Sections 4 and 5 we investigate the class qP‘) where P is the property 6 s x n + y ; we obtain a few specific characterizations in Section 4, and attempt to related the forbidden subgraphs for different values of x and y in Section 5. 2. The class F(K,+J

This is the class of graphs that do not have K,+, as a subgraph. As we shall see, it arises in many different ways. Recall that the Tur5n graph T,(n)is the complete r-partite graph with n vertices in which each vertex class has Ln/r] or [n/rl vertices. Let t,(n) be the number of edges in T,(n), and let d ( u ) denote the degree of vertex v.

Theorem 2.1. Let r 2 1 be an integer and G a graph with n vertices, m edges and minimum degree 6. Then each of the following conditions implies K,+, 5 G. (a) m >t,(n). (b) m > n2(r- 1)/2r. (c) 6 > n(r - l ) / r . (d) For some integer s, 1 S s S n, every set S of s vertices satisfies EVESd ( v ) > sn(r - l ) / r . (e) n 5 r + 1 and, for each pair of non-adjacent vertices u, u, d ( u )+ d(v) > 2n(r - l ) / r . Proof. (a) is Turfin’s theorem [5]. (b) follows because n2(r- 1)/2r L t,(n), which is most easily seen by noting that there is equality if n is divisible by r, and, between two successive multiples of r, t,(n + 1) - r,(n) is linear in n with slope 1, whereas [(n 1)’ - n2](r- 1)/2r is linear with slope (r - l ) / r < 1. (c) is special case of (d), which implies (b). To prove (e), let X : = { U EV ( G ) : d ( v ) > n(r - l ) / r } and Y:=V(G)\X, so that the induced subgraph (Y)is a complete graph. If X = 0 the result is obvious. Otherwise, let K, be the largest complete subgraph of (X). There are more than n - sn/r vertices of G, necessarily all in Y,that are adjacent to all vertices of K,, so G has a complete subgraph of order more than

+

s

+ n -sn/r

=

’r

+ (n - r)(r - s) L r r

(or else s > r). Thus K,+l E G. 0

D.R. Woodall

390 Quacterizptioa 2.2.

Let r 5 1 be an integer, Then all the following properties are subgraph-equivalent and ISG-equivalent and, for each i, q P ; ) = q P F ) = W(K,+l). Po: G # K,,,. PI: m d t,(n). pZ: rn G n2(r- 1)/2r. P3: m s f ( r , n ) , where f is any function such that f (r, n ) 3 t,(n) for all n 5 1 and f ( r , r + I) < (';I). P4:6 s n(r - l)/r. P5:6 < nr/(r+ 1). P,(s) (1 c s s r + 1, s integral and fixed): n < s or G contains some set S of s vertices such that EVES d ( v )6 sn(r - l)/r. P,(s) (1s s S r + 1, s integral and fixed): n < s or G contains some set S of s vertices such that Evesd ( v ) < snr/(r + 1). Ps: n s r or G contains two non-adjacent vertices u, v such that d ( u )+ d ( v ) s 2n(r - l)/r. Proof. It is easy to see that K,,, does not satisfy any of these conditions, so that, for each i, YP;)= %(P,")G F(K,+,). To prove the reverse inclusion we must show that a graph that fails to satisfy any of these conditions must have K,+,as an induced subgraph. This follows immediately from Theorem 2.1. 0 Chderization 2.3. (a) If 0 d x < 1 and P = P ( x ) is the property S 6 xn or the property m c jxn2, then q P C ) = %(P") = %(K,+,) where r = [1/(1- x)J. (b) Let r z = l be an integer and let the real numbers x and y s a h b

l - l / r < x < l and

(r - 1) - x r ==y < r - x ( r + 1).

If P = P(x, y ) ic. the property 6 S x n + y , then q P ' ) = W P " ) = F ( K , + J . (See Fig. 1, where each region in which q P ' ) = F ( K , + J is labelled with its forbidden subgraph Kr+l.) (c) With r, x and y ar in (b), let P be the property 2m/n e n n + y. Then qP') = cg(PE)= %(K,+*). Proof. (a) is a special case of (b) and (c). To prove (b), note that K,,, does not satisfy P since r > x ( r + 1) + y , but every graph with fewer than r 1 vertices satisfies P since t - 1s xr + y for every t =s r. We must prove that if G does not satisfy P then K,+,E G. We may suppose n = ICl > r . I f y S O the result follows from Theorem 2.l(c) since

+

S>xn+y'-

r-1-y r-1 n + y 3 y n, r

Forbidden graphs

391

Y 2

1

-

.

-1

1

0

Fig. 1. Forbidden subgraphs for

YP‘), where P is 6 S xn + y.

and if y >0 it follows likewise since

6 > x n + y >xn

3 (1- l / r ) n ,

(c) is similar to (b). 0 If v e V ( G ) and ScVCG), N ( v ) denotes the set of vertices adjacent to (“neighbouring”) v, N ( S ) : = U , , , N ( v ) and the binding number [6] of G is defined to be bind(G):=min{IN(S)I/ISI :Sc V(G), SZB, N ( S ) # V(G)}.

Theorem 2.4. Zf r 3 2 and bind(G) >y = t ( r - 1+ V

m),

then K,,, E G.

Proof. We may suppose inductively that K, E G. Suppose that no vertex of G \ K, is adjacent to all the vertices of K,. Suppose that exactly s vertices of G\K, are non-adjacent to exactly one vertex of K, and n - r - s vertices of G\ K, are non-adjacent to two or more vertices of K,. For some v in K,, at least s/r vertices of G\K, are non-adjacent to v but are adjacent to all other vertices of K,, so (if K,,, & G) together with v they form an independent set X of cardinality at least

D.R. Woodall

392

1

+ s f r . Thus n - 1 - s / r a IN(X)I > y

whence

1x1 a y ( l + s / r ) ,

r(n - 1 ) > y ( r + s ) + s.

Also, for some w in K,, at least (s + 2(n - r - s ) ) / r vertices of G\K, are non-adjacent to w. Let Y consist of these together with w itself, so that

whence

r(n - 1 ) >y(2n

- r - s).

Averaging ( 1 ) and ( 2 ) and using ( 2 ) again gives

r(n-I)>ny+$s>ny+

y ( 2 n - r ) - r ( n - 1) 2Y

,

whence

2ny2 - 2n(r - 1)y + ry

- r ( n - 1 ) < 0,

so certainly 2y2-2(r-1)y-r<0. But this contradicts the definition of y , and this contradiction completes the proof. 0

Characterization2.5. Let r L 2 be an integer and y a real number satisbing 12(r - I

+ V7T-i) s

y < r.

Let P be the property IN(S)l s y IS1for some S E V ( G )such that S # 0 and N ( S ) # V ( C ) . (3)

Then %(P') = %((P") = %(Kr+,).

Proof. It is easy to see that K r + , $% ( P ) (we need only consider I S ( = 1). It remains to prove that if G G q P ) then K,+, G G, which follows from Theorem 2.4. Problem. Let P be the property (3), where y is not in one of the ranges specified in Characterization 2.5. Then what is q P ' ) ?

In [6] I conjectured that if bind(G) 3 f then G contains a triangle ( K 3 E G). This conjecture is best possible and, if true, would solve the above problem for 3sy<2.

Forbidden graphs

393

3. The class F ( K r + l ) This is the class of graphs that do not have K,,, as a subcontraction (see [7]). We first need a lemma. For r, s 2 2 let

g(r, s):= LtrJ + LfrJ2+ *

- - + L$rJ"-',

and let g(r, 1) :=0, so that g(r, s) = L$rJ(g(r,s - 1) + 1).

Lemma 3.1.1. Let r 2 2 and let G be a graph satisfying the condition 6 > n + 1- 2n/r and such that no proper subgraph of G satisfies this condition. Let s 5 2 and suppose that G contains a vertex v such that d ( v ) 2 n - 1 - g(r, s). Then G contains some set Y of s vertices such that the induced subgraph ( Y ) is connected and V(G) \ Y c N(Y ) . Proof. Note that, if X c V(G), then some vertex of G \ X is non-adjacent to (strictly) fewer than 2 IXllr vertices of X , since G \ X does not satisfy the condition on 6. Define v l : = v andX,:=V(G)\{v,UN(v,)}. Then lXllCg(r, s), so there exists v, in G\X1, necessarily in N(vl), that is non-adjacent to fewer than 2 lXll/r, hence, to at most g(r, s - l), vertices of XI. Call the set of these X,. Since IX,l s g ( r , s - l), there exists v3 in G\X,, necessarily in N({vl, v,}), that is non-adjacent to at most g(r, s - 2) vertices of X,. Continue in this way, and set Y := {vl, v2, . . . , v,}. 0

Theorem 3.1. Let r 2 2 be an integer and let G be a graph with n vertices and minimum degree 6 > n + 1- 2n/r. Then K,,, C G. Proof. Without loss of generality, no proper subgraph of G satisfies this condition, so Lemma 3.1.1 applies. There are five cases to consider.

Case 1. n s r. Then 6 > n - 1, so this case cannot arise. Case 2. r + 1C n S !r. Then 6 > n - 2, so G is complete. Case 3. $r + t C n s 2r. Then 6 > n - 3, so G consists of t non-adjacent edges ( O s t < $ n ) . It is easy to see that by contracting Lit] edges we can form a complete graph of order n - t + Lit] a n - $t - 4 2 i n - $ 2 gr - > r ; hence, of order at least r + 1.

Case 4. n>2r and r C 5 . Then 6 > n ( r - 2 ) / r + 1 > 2 r - 3 , so 6 2 2 r - 2 and m 3 (r - 1)n. B y a theorem of Mader [3](which holds for r s 6), K,,, s G.

D.R. Woodall

394

Case 5. n > 2r and r 3 6. Choose the integer s 2 2 so that sr =sn < (s

+ 1)r. Then,

for each vertex u,

n-l-d(u)<2n/r-2<2~ and so

n - 1 - d ( u ) =s2s - 1s L$](s - 1) Q g ( r , s). By Lemma 3.1.1, there exists Y c V ( G ) such that IYI =s, (Y)is connected and V(G)\Y E N ( Y ) . The graph (V(G)\ Y) has n - s vertices and minimum degree strictly larger than

n -s

+1-2n/ra

+ 1-2(n

(n - s )

- s ) / ( r - 1).

So we may suppose inductively that K , 6 ( V ( G ) \ Y), and then K,+,Q G. 0

CbaracterizPtion 3.2. Let x and y be real numbers such that 0 G x < 1, - 1< y =z 1 and x + y a O . Let P be the property S c x n + y . Then YP<)= %(P")= Kr+ I), where

w(

r = \(I + y ) / ( l -x)J 31.

(4)

(This shows that if one draws a figure, analogous to Fig. 1, but for subcontraction-equivalence rather than subgraph-equivalence, then the regions labelled K,,, will extend up at least as far as the line y = 1.)

Proof. The definition of r implies that r 6 (1 + y)/(l - x ) < r + 1, so that r-lsrx+y

and r > ( r + l ) x + y .

(5)

Thus Kr+l is the smallest graph that does not satisfy the condition. (Compare Characterization 2.3(b).) And if G is any graph that does not satisfy the condition, with (necessarily) n > r vertices, then

G>xn+ya

r-1-y n r

r-2 +y k n + 1 =n + 1 r

~

2n/t

(6)

since y 6 1. If r k 2 then K,+l S G by Theorem 3.1, while if r = 1 then 6 > 0 implies that K,+, = K , S G . 0

Characterization 3.3. Let x and y be real numbers such that x < 1, y > 0 and + y B 0. Let P be the property

x

{ N ( S ) l < xn

Then qP') property.

+ y IS1for some S c_ V ( G )such that S # 0 and N ( S ) # V ( G ) .

= YP")= @(K,+,), where

K, is the largest complete graph with the

Forbidden graphs

395

Proof. If T := V(G)\N(S), so that T f 0, N ( T ) # V ( G ) and S E V ( G ) \ N ( T ) , it is easy to see that (N(S)IGxn+y I S l ~ $ I N ( T ) ( c y - ' ( x + y - l ) n + y - ' ( T I

and I N ( S )~~y - ' ( x +y - 1)n +y-' IS1 j JN(T)J b x n +y IT]. Thus (replacing x and y by y-'(x + y - 1) and y-' if necessary) there is no loss of generality in supposing that y S 1. The integer r satisfies (5) (and therefore (4)). We must prove that if IN(S))>xn + y (SI for every S E V ( G ) such that S # 0 and N ( S ) # V ( G ) , then K,,, c G . But this hypothesis implies (6), and K,,, G G by Theorem 3.1 if r 3 2. If r = 1 we get a contradiction to G being edgeless by taking S:= V ( G ) . 0

4. Further results involving the minimum degree

In this section and the next we shall investigate %(P'), where P = P(x, y) is the property 6 c x n + y. In this section our main aim is to justify the so-far unexplained regions in Fig. 1 . I f x + y < O t h e n K 1 $ $ ( P ) a n d s o %(P')=8. I f x + y a O a n d x b l thenall graphs satisfy P. If x < 0, then qP') and %(P) are both finite sets. If n = 0, then q P ' ) depends only on Ly] :if Ly] < 0 then %(P') is empty, if Ly] = 0 then %(P') consists of the edgeless graphs, if ly] = 1 then %(I")comprises the circuit-free graphs (forests), and so on. From Characterization 2.3(b) we know what happens if y 6 0 (see Fig. l), so from now on we assume 0 <x < 1 and y > 0. The regular trivalent graphs with n = 10 and girth 5 (the Petersen graph), and n = 14 and girth 6, show that there exist values of n and t for which the following result is best possible.

Theorem 4.1. Let t 3 4 be an integer and G a graph with n vertices and minimum degree

4:

- t)/(5t- 14) + 2 - r)/(7t - 26) + 2

(t even), (t odd).

Then G has girth Gt.

proof. The condition ensures that 6 2 2 if n < t and 6 3 3 if n a t . We may therefore suppose that 6 3 3. If G has girth >t and r is even, then all the vertices at distance 0, 1,2, . . . ,at from a given vertex are distinct, and so

n 3 1+ 6 + 6(6 - I ) + S(6 - I)*+ -

+ 6 + 6(6 - 1 ) [ 1 + ( i t - 2)2] = 1+ 6 + S(6 - l ) ( t - 3 )

31

+ 6(6 - I)+'

D.R. Woodall

3%

since 6 - 1 2 2. And if t is odd then a similar argument starting from a pair of adjacent vertices gives

n > 2 { 1 + (6 - 1) + ( 6 -I)’+ * . + ( 6 - I)*(~-’) 1 3 2 { 1 + ( 6 - 1) + ( 6 - 1)*[1+ f ( t - 5 ) 2 ] }

=2{6 + (6 - l)’(t -4)}. Suppose first that t is even. Then, since 6 3 3 and so 6’ - 6 3 56 - 9,

n - f a 1 + 6 + 6 ( 6 - l)(t - 3 ) - t 31

+ 6 + (56 - 9)(t - 3) - t

= ( 6 - 2)(5t - 14).

And if t is odd then, since 6’ - 26

n - t 3 26

+ 1 3 46 - 8 = 4( 6 - 2),

+ 2(6 - l)’(t - 4) - t

3 2(6

- 2) + 8(6 - 2)(t - 4) - t + 4

L (6 - 2)(7t - 26).

In either case we have the required contradiction.

0

Characterization4.2. Let t 3 4 be an integer and let x and y be real numbers such that (a) 2x + y 3 1, (b) tx + y C 2 , (c) ( t + 1)x + y 3 2 and

(4

- 14)x + y a 3 ((6t (8t 26)x + y 3 -

Let P be the property 6 6 xn

qP‘)

( t even), ( t odd).

+ y . Then

= qP“) = F(C3,

. . . , C,) = F(C3, . . . , C,),

the class of graphs with girth at least t

+ 1.

< 4 and y > 0. (b) ensures that C3, . . . , C,q q P ) . We must prove that if b >xn + y then G contains one of C,, . . . , C,. This is obvious if n = 1 (vacuously, since 6 2 1 ) or 2 S n G t (when 6 3 2 ) . If t + 1 s n < 6t - 14 and t is even, then (c) implies 6 2 3 and Theorem 4.1 implies Ppoof. (a), (b) and (c) imply O < x

the result, while if n 3 6t - 14 and t is even then (b) and (d) imply (5t - 14)6 > (51 - 14)(xn

+ y ) = [(6t - 14)x + y ] ( n - t ) - (tx + y ) [ n - (6t - 14)] 3 3(n

- t ) - 2[n - (6t - 14)]

=n -t

+ 2(5t - 14)

and the result again follows from Theorem 4.1. The result for odd t follows similarly. 0

Forbidden graphs

397

Corollary 4.2.1. Zf t a 4 and P is any of the properties n n-2 n-1 +1, 6 s + 1 and 6<--+1, t-1 t t+l

6s-

then %(P') is as above. The following theorem is best possible when n = 4t - 2 in view of the graph KU-l,2f-lminus a 1-factor. When t = 4 or 3, we shall improve the result slightly for larger values of n in Theorems 4.5 and 4.7 respectively.

Theorem 4.3. Zf t a 3 and G is a graph such that 2t6 > (t - l ) ( n + 2), then G has a K3 or a Kt,tas a (necessarily induced) subgraph. Proof. Suppose not. Choose adjacent vertices u, v in G and sets U ~ N ( u ) \ { v } and V =N(v)\{u} such that lUl= IVl= 6 - 1. Since G is triangle-free, U and V are independent sets and U n V = 0. Let W := V(G)\ ({u, v ) U U U V),so that 0 s IWl = n -26 = n

2t6 --+t-1

26 t-1

26 t-1

< n - (n + 2) + -by hypothesis n

L

=-(6

t-1

-t

+ 1 ) s 6 - t + 1.

Hence IWl s 6 - t. Choose a in U U V such that IN(a) n WI is maximal. Without loss of generality a E U and (since IWI d 6 - t ) a is adjacent to vertices b l , . . . , b,-l in V. For each i, N(a) n N(bJ = 0, and so

IN(bi)flWI<min((N(a)flWI, I W \ N ( a ) l ) q $ J W ( = $ n - 6 , whence bi is non-adjacent to at most f n - 6 vertices of U. Since all but at most t - 2 vertices of U are non-adjacent to some bi (otherwise Kt-l,t-l E (UU V j and Kt,t

= GI,

( 6 - I ) - (t - 2) < (t - I)(& - 6) and so 2t6 s (t - l ) ( n + 2), contrary to hypothesis. 0

Characterization 4.4. Let t k 5 be an integer and let x and y be real numbers such that (a) 3x+y<2, (b) 2tx+y
Proof. (a) and (b) ensure that K3 and Kt,t$ %(P). (a) and (c) imply that (2t - 5)x > t - 3, whence 2tx>-

2t(t - 3) at-1 2t - 5

(7)

D.R. Woodoll

398

since t

> 0. If n s 2(t - 1 ) and G $ q P ) , then (c)

5. (b) and (c) imply that y

implies that

6 >xn

Yn + y 2 x n + -2 2(t - 1 )

n, 2

so that K 3 c G. And if n > 2(t - 1 ) and G $ qP),then (c) and (7) imply that 2 6 > 2 ( x n + y ) = 2t[2(t - l)x + y ] + %[n - 2(t - l ) ] > 2t(t - 1 ) + ( t - l ) [ n - 2(t - l ) ] = (t - l)(n 50

+ 2),

that G contains K 3 or K,,, by Theorem 4.3. 0

TBeorem 4.5. If G is a graph such that 6 > f n + 1 then G has a K 3 or a K4,4as an induced subgraph unless G is K7,7 minus a 1-factor. Proof. Suppose not. Define u, u, U, V, W and a as in Theorem 4.3, let s := IN(a) n Wl and note that, since n C 36 - 4, s ~lWl = n -26 d 6 -4.

(Hence 6 a 4). There are at least 6 - 1 - s L 3 vertices of V that are adjacent to a, and each of them is adjacent to at most min(s, IWl - s) vertices of W and hence to at least max(6 - 1 - s, 3 + s) vertices of U. So some two of the 6 - 2 vertices in U\{a} are adjacent to the same three vertices of N ( a ) i l V (giving a Kdn4in G) unless both

(6-1-s)(6-;-s)s2(

6-2

)

and

(6-1-s)(

2+s

)s2(

6-2

).

(9)

Now,the LHS of (8) decreases with s (over the range of interest), and the LHS of (9) is an “upside-down” cubic with zeros at -2, - 1 and 6 - 1. Since (9) is obviously false if s takes its maximum value of 6 - 4,it follows that if (8) and (9) hold then they must both do so when 6 - 2 - s = 2 + s. This gives

which implies (?a + l ) f 6 G 4(6 - 3), whence 6 S 6 =s8. Trying values in (8) and (9), we find that the only possibilities are 6 = 6, s = 1 and 6 = 8, s = 2. In each case there is equality in both (8) and (9), so IW (= 2s and n = 26 + 2s = 14 or 20 respectively. It is easy to see that all vertices of U are adjacent to the same s vertices of W and all vertices of V are adjacent to the other s vertices of W. Thus

Forbidden graphs

399

in the first case G is K7,7minus a 1-factor, and the second case is impossible since G must contain K4,4. 0

Characterization 4.6. Let x and y be real numbers such that (a) 3x + y < 2, (b) & r + y < 4 a n d ( c ) & r + y 3 3 . LetPbetheproperty 6 d x n + y . Zf(d) 1 4 x + y < 6 , then q P ' ) = %((P") is the class of graphs that do not contain K3, K4,4or K,,7 minus a 1-factor, while if 14x + y 3 6 then %(I"=) = %(P") = W ( K 3 ,K4,4).

Proof. (a) and (b) ensure that K 3 and K4,44 q P ) , while (d) does the same for K7,7 minus a 1-factor. (b) and (c) imply that y > 0. If n d 6 and G 4 q P ) , then (c) implies that

6 >xn

n + y Z x n +Y-36 2'

so that K3 G G. And if n 3 6 and G $ %(P) then (a) and (c) imply that 36 > 3(xn + y) = (6.x + y)(n - 3) - (3x + y)(n - 6) 3 3 ( n - 3) - 2(n - 6) = n + 3, so the result follows from Theorem 4.5.

0

Theorem 4.7. If G h a graph such that 6 > $(n + 4) then G has a K 3 or a K3,3as an induced subgraph.

Proof. This follows from Theorem 4.3 if n d 10, so suppose n 3 11, 6 2 5. Suppose G does not contain K3 or K3,3.It must contain K2,3, since C , E G by Theorem 4.1, and if no vertex of G \ C4 is adjacent to more than one vertex of C4 then G \ C4satisfies the hypothesis and so K2,3 c G \ C4 by induction. Define u, v, U,V , W as in Theorem 4.3 and let

x:={ X E u u v: I N ( X n) ( u u v)l32}; since K2,31~G we can choose u, v, U, V so that X # 0. Choose a E X such that s := (N(a)f l WJ is maximal. Note that n d 346 - 44, so IWl= n - 26 d 1$6 - 44. Suppose w.1.o.g. a E U.There are two cases to consider.

Case 1. N ( a ) n (V\X) # 0. Let b belong to this set. Since N ( a ) nN ( b ) = 0 and IN(b) n Wl s 6 - 2, it follows that s + S - 2 s IWI d 146 - 44 and s d fS - 2i. Thus JN(a)n VI 3 6 - 1- s 3 $6 + 14. For each b' in N ( a ) rl X , IN(b') n U l 2 $6 + 14 similarly (by the maximality of s), and so IN(a) nX(d 1 or else K2,2 (UU V) and K3,3c G. So IN(a) n (V\X)l 3 46 + 4 3 3, and any three of these vertices form a K3,3with W since they are all adjacent to at least 6 - 2 verticesof Wand 3[lWl-(6-2)]dIWl-3. Case 2. N ( a ) n (V\X) = 0. Then there are at least max(2, 6 - 1- s) vertices of V that are adjacent to a, and each of them is adjacent to at most

D.R. Woodall

400

+

min(s, 1 W J- s) vertices of W and hence to at least max(6 - 1- s, s - 46 34) vertices of U. So some vertex in U\{a} is adjacent to two vertices of N ( o ) n V (giving a K3,3in C ) unless max(2, 6 - 1-s)max(b

- 2 -s,

s - 46 +24)

s IU\{a}l= 6 -2;

this implies

(6 - 1 -s)(6

- 2 - s ) =s6

and rnax(2, 6 - 1 - s)(s

-2

- 46 + 2 t ) c 6 - 2.

(11) is obviously false if s 2 S - 3, and so if (10) and (11) hold then they must both do so when 6 - 2 - s = s - 46 +2$, s = $6 - $. This gives

+ 5)+(S + 1) s 6 - 2 or h2 - 106 + 37 S 0, which is impossible. i(6

Characterization 4.8. Let x and y be real number such that (a) 3x + y < 2, (b) 6 x + y < 3 , (c) 4 x + y 2 2 a n d ( d ) 1 0 x + y 3 4 . LetPbetheproperty 6 a x n + y . Then q P ' ) = q P " ) = W ( K 3K3,3). , (Note that graphs such as 2oC5 and K5,s minus a 1-factor, which have n = 10, 6 = 4 and no K 3 or K3,3, show that (d) correctly delimits the region labelled K 3 , K3,3in Fig. 1. Here 2oG is the graph obtained from G by doubling each vertex and replacing each edge by a K2,*.)

Proof. (a) and (b) ensure that K 3 and K3,3$ q P ) . (b) and (c) imply that y > 0. If n c 4 and G $ q P ) , then (c) implies that Yn n 6>xn+y~xn+-2-, 4 2 so that K 3 c G. If 4 G n C 10 and G @ q P ) then (c) and (d) imply that 66>6(xn + y ) = ( 4 x + y ) ( l O - n ) + ( l O x + y ) ( n - 4 ) 3 2(10 - n )

+ 4(n - 4) = 2(n + 2),

so that G contains a K 3 or a K3,3 by Theorem 4.3. And if n 3 10 and G $ %(P) then (a) and (d) imply that

76 > 7(xn + y) = (31 + y)(10 - n ) + (lox + y)(n - 3) 2 2(10 - n ) + 4(n - 3) = 2(n + 4).

so the result follows from Theorem 4.7. I7 We write K,(r) for the Turan graph T,(rt), the complete r-partite graph on r sets of t vertices each. The following theorem makes no pretence of being best possible-indeed, if r = 2 it is worse than Theorem 4.3 whenever n > 2r - 2-but it is simple to prove and enables us to fill in the depth of the notches between the K,,, regions in Fig. 1.

Forbidden graphs

401

Theorem 4.9. Let r 3 2 and t 3 1 be integers and G a graph with n vertices and minimum degree 6 > n - (tn - t + l ) / ( r t- 1). Then K,,, or K,(t) E G. Proof. We prove the result by induction on t, noting that it holds if t = 1 (when the conclusion is that K , E G) by Theorem 2.l(c). So suppose that t 3 2 and neither K,,, nor K,(t) E G. We may suppose inductively that K,(t - 1)E G. Let G, = T,(nl) be the largest r-partite Turfin subgraph of G (necessarily induced), where n l = r ( t - l ) + s and O S s < r . Note that G1#G, since if n = n l a n d s = O then d 1 : = 6 ( G 1 ) = n l - t + 1 whereas 6 > n - t + l , and if l S s < r then a,= n1- t whereas 6>n-t

+ l-- rt ts- 1 > n - t .

There is no loss of generality in supposing that G \ G1 does not satisfy the degree condition, so that at least one vertex of G\Gl is non-adjacent to fewer than t n , / ( r t - l ) S t , hence to at most t - 1, vertices of G,. But this means that G contains a K,,, or a T,(n, l ) ,contrary to hypothesis. 0

+

Characteriza~on4.10. Let r > 2 and t 3 1 be integers and x and y real numbers such that (a) rtx + y < (r - l)t, (b) r(t - 1)x + y 2 (r - l)(t - 1) and (c) (rt - 1)x > rt - t - 1. Let P be the property 6 S xn + y. Then %(P') = %(P") = v ( K r + 1 , Kr(t)).

Proof. (a) and (b) imply that rx < r - 1 and y > 0. (a) and (c) imply x + y < 1 and so (with the previous sentence) ( r + 1)x + y < r, whence K,,, @ %(P). (a) ensures that K,(t) @ %(P). If n S r(t - 1) and G @ %(P) then (b) implies 6>xn+yaxn+-

yn n ( r - 1) 2 , r ( t - 1) r

so that K,,, G G by Theorem 2.l(c). And if n >r(t - 1) and G @ %(P) then (b) and (c) imply

(rt - 1)6 > (rt - l ) ( x n + y ) = (rt - l)[r(t- l ) x + y ] + (rt - l ) x [ n- r(t - l ) ] 2 (rt - l ) ( r - l ) ( t - 1) + [rt - t - l ] [ n- r(t - l ) ] = (rt - 1)n - (tn - t

+ l),

so that G contains K,,, or K,(t) as a subgraph by Theorem 4.9.

0

5. The shift theorems In this section we attempt to relate the forbidden subgraphs for the property 6 ~ x + ny for different values of x and y. If P = P(x, y ) denotes this property, let

D.R. Wmdall

402

-P denote the property 6 > x n + y . Let S c ( x , y ) be the set of graphs in q - P ) that have no proper subgraph in Y-P), and let P ( x , y ) be the set of graphs in q - P ) that have no proper induced subgraph in Y - P ) , so that S c ( x , y ) c P ( x , y ) and

%(F) = %P")

= W(SC(X,y ) ) = W(9"(x,

y ) )= W ( P ( x , y)).

In all the examples we have seen so far, it has been the case that F ( x , y ) = F ( x , y ) , and all the graphs in this set are regular. Lest the reader presume that this is true in general, we should immediately point out a consequence of Corollary 4.2.1 and Theorem 5.2 (below), that (for example) C5+ K2E F(:2). , (Here denotes "join": every vertex of C5 is adjacent to every vertex of E2.) Also C5+ K 2 E . P ( a , 2), since this graph has the same minimum degree and number of vertices as C5+ K2,and it is easy to check that every induced subgraph of it satisfies 6 ss $n + 2. Thus the irregular graph C5+ R2 is in both 2 ) and Sc(a,2 ) , while C5+ K2is in the second set only. If x f 2 , define the ship of ( x , y ) to be

"+"

$-(a,

S fixes the line x = 1, and on any other line through (1,O) the segment between 0 and 4 is mapped onto the segment between $ and 3, which is mapped onto the segment between 3 and f , and so on. Thus in Fig. 1 the region labelled K2 is mapped by S onto the region labelled Kf, which is mapped onto the region labelled K.,,and so on. If G is a graph with n vertices and minimum degree 6, define the shift S(G) of G to be G K,,-b; it has 2n - 6 vertices and minimum degree n. From Fig. 1 we see that the following conjecture is true if x + 2y s 1. The dotted lines in that figure between x = 1 and x = $ are the shifts of the continuous lines between x = 0 and x = 4, and so represent conjectures whose truth is implied by the following:

+

Conjecture (The Shift Conjecture). If 0 =sx s 1 and y b -1, then Sc(S(x, y ) ) = {S(G):G E F ( x , y)} and S"(S(x, y ) ) = { S ( G ) :G

E

Sc(x, y ) } .

w r n 5.1. l f O s x 6 l a n d y a - 1 , then { S ( G ) : G E F ( X , ~ ) } E ~ ~ ( S ( X , ~ ) ) and { S ( G ) : GE 9"(x, Y ) ) c 9 c ( S ( x , Y ) ) .

Proof. Note first that

so that G satisfies P ( x , y) if and only if S ( G ) satisfies P ( S ( x , y ) ) . Suppose now that G is a graph in 9"(x, y ) , so that G does not satisfy P ( x , y )

Forbidden graphs

403

but every proper induced subgraph of it does; in particular,

6 - 1 d x ( n - 1) + y .

(12)

By the previous paragraph, S(G) does not satisfy P(S(x, y)). We must prove that every proper induced subgraph of it does. Such an induced subgraph is either G1+ Kr,where Gl is a proper induced subgraph of G and r d n - 6, or G + Kr where r d n - 6 - 1. The latter graph has n + r vertices and minimum degree 6 + r and satisfies P(S(x, y)) since (12) is equivalent to (2 - x ) ( n

- 1 ) c 2 n - 6 - 1 +y,

(13)

which implies (2 - x ) ( 6

+r)

(n + r ) + y

because 6 + r d n - 1 and 2 - x degree S1, then

2 1.

And if Gl has n1 vertices and minimum

d1c x n l + y and (2 - x ) n l c2n1- + y (14) since G1 satisfies P, and Gl+Kr has n l + r vertices and minimum degree

min(nl, S1+ r). In either case the required inequality follows immediately from (14)This proves the second inclusion in the statement of the theorem. To prove the first, suppose that G is a graph in S c ( x , y) E 9 " ( x , y). We must prove that every proper subgraph of S(G) satisfies P(S(x, y)). It suffices to consider subgraphs of the same form as before, where now G1 is a proper subgraph of G (not necessarily induced). The only difference in the proof arises if r = n - 6 and n1= n. But then 61C 6, since every proper subgraph of G satisfies P(x, y) and G does not, and since Gl + Kr has minimum degree at most 6 - 1 + r, the result follows from (13). 0 Finally, there is an upwards shift theorem, although it is less interesting as equality frequently does not hold. Define U(x, y) := (x, y + 1- x ) and let U ( G ) be obtained from G by adding a single vertex adjacent to all vertices of minimum degree in G. Theorem 5.2. { U(G):G E F ( x , y ) } c Sc( U(x, y ) ) .

Proof. Let G be a graph in S c ( x , y ) with n vertices and minimum degree 6. Since G does not satisfy P(x, y) and every proper subgraph of it does, deleting any edge of G must lower its minimum degree. Thus no edge of G joins two vertices with degree greater than 6. It follows immediately that G has at least S + 1 vertices of degree 6, so that U(G) has minimum degree 6 + 1. U ( G )clearly has n + 1 vertices. Since 6 > x n + y it follows that 6 + 1 > x ( n + 1) + ( y + 1 - x ) , so that U(G)does not satisfy P ( U ( x , y ) ) . The deletion of any edge from U ( G )

404

D.R. WoodaU

results in a graph with minimum degree 6, which satisfies P ( U ( x , y)) since 6 - 1s xn + y. So it suffices to consider induced subgraphs of U ( G ) obtained by deleting vertices of G , which will have n , + 1 vertices and minimum degree at most d1 + 1 where 6, <xnl +y, and which therefore clearly satisfy P ( W Y ))*

References [l] L.W.Beineke, Characterizations of derived graphs, J. Combinat. Theory 9 (1970) 129-135. f2] L.Lovisz, A characterization of perfect graphs, J. Combinat. Theory Ser. B 13 (1972) 95-98. 131 W.Mader, Homomorphiesatze fiir Graphen, Math. Ann. 178 (1968) 154-168. [4] H. Meyniel, On the perfect graph conjecture, Discrete Math. 16 (1976) 339-342. [5] P. Turin, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436-452. [6] D.R. Woodall, The binding number of a graph and its Anderson number, J. Combinat. Theory Ser. B 15 (1973) 225-255. (71 D.R. Woodall, Subcontraction-equivalence and Hadwiger’s conjecture, J. Graph Theory 11 (1987) 197-204.

Discrete Mathematics 75 (1989) 405-407 North-Holland

COMBINATORICS 1988 LIST OF CONTRIBUTORS J. Akiyama

I. Algor

Noga Alon

J.-C. Bermond

K. Berrada

Norman Biggs

BCla Bollobhs

J. Bond

Adrian Bondy

Endre Boros Graham Brightwell

Peter J. Cameron Amanda G. Chetwynd

Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-12, Japan Department of Mathematics, Sackler Faculty, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Department of Mathematics, Sackler Faculty, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Laboratoire de Recherche en Informatique, UniversitC Paris-Sud, Bit. 490, 91405 Orsay Cedex, France Laboratoire de Recherche en Informatique, UniversitC Paris-Sud, Bit. 490, 91405 Orsay Cedex, France Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, England Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Laboratoire de Recherche en Informatique, UniversitC Pas-Sud, Bit. 490, 91405 Orsay Cedex, France Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 RUTCOR, Department af Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, EngZand School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England Department of Mathematics, University of Lancaster, Cartmel College, Bailrigg, Lancaster LA1 4YL, England

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

c o m b ~ o r i c s1988

406

F.R.K. Chung Michel Deza Reinhard Diestel

Paul ErdEis

Phillippe Flajolet Zoltb Fiiredi

H. Furstenberg Roland Haggkvist AndrL Hajnal

Peter L. Hammer Julie Haviland

A.J.W. Hilton

Y.Katznelson Donald E. Knuth Monique Laurent

Imre Leader

Bell Communications Research, Morristown, NJ 07%9, USA CNRS, UA 212, UniversitC Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Mathematical Institute of the Hungarian Academy of Sciences, ReAitanoda utca 13-15, 1053 Budapest V., Hungary INRIA, Rocquencourt, 78150 Le Chesnay, France Mathematical Institute of the Hungarian Academy of Sciences, 1364, Budapest, P.O. Box 127, Hungary Department of Mathematics, The Hebrew University, Jerusalem, Israel Matematiska Institutionen, Stockholms Universitet, Box 6701, 113 85 Stockholm, Sweden Mathematical Institute of the Hungarian Academy of Sciences, Rehltanoda utca 13-15, 1053 Budapest V., Hungary RUTCOR, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Department of Mathematics, University of Reading, Reading, P.O. Box 220, Whiteknights, Reading RG6 2AX, England Department of Mathematics, Stanford University, Stanford, CA 94305, USA Computer Science Department, Stanford University, Stanford, CA 94305, USA CNRS, Lamsade, Universitd Paris Dauphine, Place du MarCchal de Lattre de Tassigny, 75775 Paris Cedex 16, France Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Department of Computer Science, Eotvos University, Budapest, Hungary and Department of Computer Science, Princeton University, Princeton, NJ 08544,

USA

List of Contributors

Fumi Nakada Jaroslav NeSetfil

J.L. Nicolas

Boris Pittel Steen Rasmussen

Vojtgch Rod1 P. Rosenstiehl Michael Saks

407

Department of Mathematics, Tokai University, Hiratsuka, Kanaga 259-12, Japan Department of Applied Mathematics, Charles University, Malostranskk n6m. 25, 118 00 Prague 1, Czechoslovakia Groupe Logiques, Mathkmatiques Discr&tes,Informatique Institute des Sciences et de la Matihe, Universitk Lyon 1,/bd. du 11 Novembre 1918, 69622 Villeubanne Cedex, France Mathematics Department, Ohio State University, Columbus, OH 43210, USA Center for Nonlinear Studies and Theoretical Division, MS B258, LANL, Los Alamos, NM 87545, USA and Physics Laboratory 111, The Technical University of Denmark, 2800 Lyngby, Denmark Department of Mathematics, Czech Technical University, Prague, Czechoslovakia Centre d'Analyse et de Mathkmatique Sociale, 54, Boulevard Raspail, 75006 Paris, France RUTCOR, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA and Bell Communications Research, Morristown, NJ 07960,

USA A. Skkozy

Paul D. Seymour L.A. Szkkely Andrew Thomason

Sinichi Tokunaga W.T. Trotter D. R. Woodall

N .C. Wormald

A Magyar Tudomhyos Akadkmia, Matematikai Kutato Intkzete, Realtanoda u. 13-15, Pf. 127, H1364 Budapest, Hungary Bell Communications Research, Morristown, NJ 07960, USA ELTE Matematikai Intkzet, Mdzeum krt 6-8, Budapest 1088, Hungary Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 lSB, England Tokyo University, Hongo, Bunkyo-ku, Tokyo, 113, Japan Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, England Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand

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Discrete Mathematics 75 (1989) 409-410 North-Holland

409

AUTHOR INDEX Volume 75 (1989) Akiyama, J., F. Nakada and S. Tokunaga, Packing smaller graphs into a graph Algor, I. and N. Alon, The star arboricity of graphs Alon, N., see Algor, I. Alon, N. and B. BollobBs, Graphs with a small number of distinct induced subgraphs Bermond, J.-C., K. Berrada and J. Bond, Extensions of networks with given diameter Berrada, K., see Bermond, J.-C. Biggs, N., Confluence of some presentations associated with graphs Bollobtk, B., Preface Bollobi, B., Paul Erdiis at Seventy-Five BollobBs, B., see Alon, N. BollobBs, B. and G. Brightwell, Long cycles in graphs with no subgraphs of minimal degree 3 Bollobiis, B. and S. Rasmussen, First cycles in random directed graph processes Bond, J., see Bermond, J.-C. Bondy, A., Trigraphs Boros, E. and P.L. Hammer, On clustering problems with connected optima in Euclidean spaces Brightwell, G., see Bollobiis, B.

(1-3) 7-9 (1-3) 11-22 (1-3) 11-22 (1-3) 23-30

(1-3) 31-40 (1-3) 31-40 (1-3) 41-46 (1-3) 1 (1-3) 3-5 (1-3) 23-30 (1-3) (1-3) (1-3) (1-3)

47-53 55-68 31-40 69-79

(1-3) 81-88 (1-3) 47-53

Cameron, P.J., Some sequences of integers Chetwynd, A.G. and A.J.W. Hilton, 1-Factorizingregular graphs of high degreean improved bound Chung, F.R.K. and P.D. Seymour, Graphs with small bandwidth and cutwidth

(1-3) 103-112 (1-3) 113-119

Deza, M., see Laurent, M. Diestel, R., Simplicia1 decompositions of graphs: A survey of applications

(1-3) 279-313 (1-3) 121-144

Erdiis, P. and A. Hajnal, On the number of distinct induced subgraphs of a graph Erdiis, P., J.L. Nicolas and A. S6rkiizy, On the number of partitions of n without a given subsum (I)

(1-3) 145-154

Flajolet, P., D.E. Knuth aand B. Pittel, The first cycles in an evolving graph Fiiredi, Z.,Covering the complete graph by partitions Furstenberg, H. and Y. Katznelson, A density version of the Hales-Jewett theorem fork=3

(1-3) 167-215 (1-3) 217-226

Haggkvist, R., On the path-complete bipartite Ramsey number Haggkvist, R., Towards a solution of the Dinitz problem? Haggkvist, R., A note on Latin squares with restricted support Hajnal, A., see Erdiis, P. Hammer, P.L.,see Boros, E. Haviland. J. and A. Thomason, Pseudo-random hyDergraDhs -_ Hilton, A.J. W., see Chetwynd, A.G.

(1-3) 243-245 (1-3) 247-251 (1-3) 253-254 (1-3) 145-154 (1-3) 81-88 (1-3) 255-278 (i-3j 103-112

I

-

(1-3) 89-102

(1-3) 155-166

(1-3) 227-241

410

A h r index

Katznetson, Y.,see Furstenberg. H. Knuth, D.E., see Flajolet, P.

(1-3) 227-241 (1-3) 167-215

Laurent. M. and M. Deza, Bouquets of geometric lattices: Some algebraic and topologicai aspects Leader, I., A short proof of a theorem of V h o s on matroid representations LovBsz, L., M.Saks and W.T. Trotter, An on-line graph coloring algorithm with sublinear performance ratio

(1-3) 279-313 (1-3) 315-317

Nakada, F., see Akiyama, J. Nektfil, J. and V. R M , The Partite construction and Ramsey set systems Nicdas, J.L.,see Erdiis, P.

(1-3) 7-9 (1-3) 327-334 (1-3) 155-166

Pittel, B.,see Flajolet, P

(1-3) 167-215

Rasmussen, S., see BoUob& B.

(1-3) 55-68 (1-3) 327-334 (1-3) 335-342

Riidl, V.,see NeSetKl, J. Roscnstiehl, P., Scaffold permutations

Saks, M.,see h v k , L. Sdrkazy, A., see Erdiis. P. Seymour. P.D., see Chung, F.R.K. SzCkeIy, L.A. and N.C. Wormald. Bounds on the measurable chromatic number of R"

Tbomason, A., A simple linear expected time algorithm for finding a hamilton path Thomason, A., Dense expanders and pseudo-random bipartite graphs Thomason, A., see Haviland, J. Tokmaga, S., see Akiyama, J. Trotter, W.T., see L o v k L.

W&,

D.R., Forbidden graphs for degree and neighbourhood conditions Wormaid, N.C., see Sdkely, L.A.

(1-3) 319-325

(1-3) 319-325 (1-3) 155-166 (1-3) 113-119 (1-3) 343-372 (1-3) 373-379 (1-3) 381-386 (1-3) 255-278 (1-3) 7-9 (1-3) 319-325 (1-3) 387-404 (1-3) 343-372