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Every H -decomposition of K n has a nearly resolvable alternative

Every H -decomposition of K n has a nearly resolvable alternative. Wilson: e ( H ) | n ( n -1)/2 and gcd ( H ) | n -1 n >> then there exists an H -decomposition of K n . There may be many distinct H -decompositions of K n which vary in their properties.

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Every H -decomposition of K n has a nearly resolvable alternative

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  1. Every H-decomposition of Kn has a nearly resolvable alternative • Wilson: e(H) | n(n-1)/2 and gcd(H) | n-1 n>> then there exists an H-decomposition of Kn. • There may be many distinct H-decompositions of Kn which vary in their properties. • Example: H=Kkk | n k-1 | n-1 n>> then there exists a resolvableKk dec. of Kn. This is a theorem of Ray-Chaudhuri & Wilson. There are also non-resolvable ones. • There is no analog of the theorem of RC & W for general graphs H. In fact it is not true for some H. (e.g. H=K1,t where t > 2 is odd). • The resolution numberc(H,n):let L be an H-dec. of Kn. c(L) is the chromatic number of the intersection graph of L.c(H,n) = minL c(L).

  2. By RC & W c(Kk,n) = (n-1)/(k-1) iffn=kmod k(k-1) and n >>. • Trivially,c(H,n) ³ (n-1)h/(2m) where m=|e(H)|. Equality holds iff there is a resolvable H-decomposition. The main result • Let H be a fixed graph with h vertices and m edges. Then: c(H,n) = (1+o(1)) (n-1)h/(2m). • The o(n) term is, in fact, of the form nb where b < 1. The error term cannot be omitted. Outline of proof • First, we show that if Kn is H-dec. then it can also be decomposed into H-decomposable cliques whose sizes are bounded. • This follows from a theorem of Wilson regarding pairwise balanced designs, together with an additional simple set-theoretic argument. • We also need to use the powerful theorem of Pippenger & Spencer regarding the chromatic index of uniform hypergraphs:

  3. Let h and C be positive integers and let a < 1 and e< 1 be positive reals. There exists N0=N0(h,C,a,e) and 0 < b= b (h,C,a,e) < 1 such that the following holds: If S is an h-uniform hypergraph with n>N0 vertices and: • There exists d > en such that for every vertex x |deg(x)-d| < da. • Any two vertices appear together in at most C edges. Then, q(S) < d+db. • Every H-dec. defines an h-uniform hypergraph whose edges correspond to the vertices of each member of the decomposition. Clearly, the chromatic index of this hypergraph is what we need to bound. We need to show there is an H-dec. whose hypergraph satisfies the conditions of P & S with d=(n-1)h/(2m). • We need the following large deviation result: • For every a>0 there exists t=t(a) such that if t >T and X1 , . . . , Xt are t mutually independent discrete r.v. taking values between 0 and a and m is the expectation of X= X1 + . . . + Xt then:prob[|X- m| > t0.51] < t -2. Proof is a simple use of Azuma’s inequality.

  4. Combining it all together • Let F=F(H)be a finite set of integers with the property that if Kn is H-dec. then Kn is also decomposable into H-decomposablecliques whose sizes belong to F. Define C=(k-1)/d(H) where k is the largest integer in F. Define e=h/(3m) < 1. Let a=0.6, and let b and N0 be as in P & S. • For each f  F let Lfbe a fixed H-dec. of Kf . let Yf be the r.v. corresponding to the number of members of Kf containing a randomly selected vertex. Note that Yf is discrete and0 < Yf < k. • We show that if n >> and Knis H-decomposable then c(H,n) < d+dbwhered=(n-1)h/(2m). • Let L* be a decomposition of Kn intoH-decomposable cliques whose sizes belong to F. Each Q  L* is isomorphic to some Kf so there are f! different ways to decompose Q into copies of H using Lf . for each Q  L* we randomly and uniformly choose such a permutation. All |L*| choices are independent. This defines a random H-dec. of Kn denoted L.

  5. We show that with positive probability, each vertex of Kn appears in at least d-da and in at most d+da members of L. This follows (with a little work) from the large deviation lemma. • Any two members of Kn appear together in at most C members of L. • The last two claims show that P & S holds for the hypergraph corresponding to L, with positive probability. A conjecture • The o(n) error term in the result can be replaced by a constant (which depends only on H). Namely: c(H,n) =(n-1)h/(2m) +C(H). Another small goodie • For every H there are infinitely many n, for which there exists an H-dec. L of Kn such that the intersection graph of L is regular of degree (n-1)h/(2m). (This is only interesting if H is not a regular graph).

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