Fractional cycle decompositions in hypergraphs

Abstract

We prove that for any integer k ≥ 2 and 𝜀 > 0, there is an integer 𝓁 0 ≥ 1 such that any k-uniform hypergraph on n vertices with minimum codegree at least (1∕2 + 𝜀)n has a fractional decomposition into (tight) cycles of length 𝓁 (𝓁-cycles for short) whenever 𝓁 ≥ 𝓁 0 and n is large in terms of 𝓁. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into 𝓁-cycles. Moreover, for graphs this even guarantees integral decompositions into 𝓁-cycles and solves a problem posed by Glock, Kühn, and Osthus. For our proof, we introduce a new method for finding a set of 𝓁-cycles such that every edge is contained in roughly the same number of 𝓁-cycles from this set by exploiting that certain Markov chains are rapidly mixing.

Cycle decompositions in $k$-uniform hypergraphs

“…The next lemma finds many trails of prescribed length which connect any given pair of ordered pk ´1q-tuples. It follows from [11,Lemma 2.3]. Lemma 6.1.…”

“…Here, we give lower bounds on the parameter δ Cpkq . Joos and Kühn [11] showed that δ Cpkq ě 1 2 `1 pk´1`2{kqp ´1q holds for each k ě 2 and not divisible by k. We give new bounds, which remove the dependency on k.…”

“…The case k " 3 is proven by the last two authors in [14] and here we follow the same lines. Given a k-graph H and an edge e P H, recall that C pHq and C pH, eq are the family of all -cycles in H and those containing e. The proof of Lemma 9.5 rests in a result by Joos and Kühn [11] about fractional C k -decompositions. Theorem 9.6 (Joos and Kühn [11]).…”

“…Our proof of the Cover-down lemma requires a result of Joos and Kühn on 'fractional decompositions' [11], and a detour which finds and uses (tight) path decompositions.…”

“…For odd values of , the situation is different. Joos and Kühn [11] showed that δ C p2q " 1{2 `c , where c is a sequence of non-zero numbers depending on only, which satisfy c Ñ 0 when Ñ 8.…”