Cycle decompositions in $3$-uniform hypergraphs

Abstract

We show that 3-graphs on n vertices whose codegree is at least p2{3 `op1qqn can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to the op1q term. All together, our results answer in the negative some recent questions of Glock, Joos, Kühn, and Osthus.

Fractional cycle decompositions in hypergraphs

“…This in particular includes the case that 𝛿(H) ≥ (1∕2 + o(1))n, settles a question posed by Glock, Kühn, and Osthus for graphs [5], and improves significantly on previous results [4]. This is particularly interesting as very recently, Piga and Sanhueza-Matamala [11] showed that for 3-graphs the decomposition threshold for (long) cycles is in fact 2∕3. It is therefore plausible that for hypergraphs of higher uniformity this threshold is also significantly higher than 1∕2.…”

High-Girth Steiner Triple Systems

“…This is analogous to the fact that for fixed j ∈ N, in a random triple system with m = N 2 triples, the number of (j, j − 2)-configurations is of the same order m, but in a random triple system with m = o(N 2 ) triples, the number of (j, j − 2)configurations is negligible compared to m. Actually, this is the key fact underlying Brown, Erdős, and Sós' lower bound for the (j, j − 2)-problem 6. Related ideas using Euler tours also appear in papers of Keevash[35] and Piga and Sanhueza-Matamala[50].…”

Moderate Deviations of Triangle Counts in Sparse Random Graphs

“…The next result compares probabilities of sum of Bernoulli random variables (not necessarily independent) with binomial random variables. This seems a fairly standard result that may be found in [25], for example. Lemma 2.17.…”