Tight cycles in hypergraphs

We apply a recent version of the Strong Hypergraph Regularity Lemma(see [1], [2]) to prove two new results on tight cycles in k-uniform hypergraphs. The first result is an extension of the Erdos-Gallai Theorem for graphs: For every > 0, every sufficiently large k-uniform hypergraph on n vertices with at least edges contains a tight cycle of length @n for any @ 2 [0; 1]. Our second result concerns k-partite k-uniform hypergraphs with partition classes of size n and for each @ 2 (0; 1) provides an asymptotically optimal minimum codegree requirement for the hypergraph to contain a cycle of length @kn

The Existence of Hamilton Cycle in n-Balanced k-Partite Graphs

Let $G_{k,n}$ be the $n$-balanced $k$-partite graph, whose vertex set can be partitioned into $k$ parts, each has $n$ vertices. In this paper, we prove that if $k \geq 2,n \geq 1$, for the edge set $E(G)$ of $G_{k,n}$ $$|E(G)| \geq\left\{\begin{array}{cc} 1 & \text { if } k=2, n=1 n^{2} C_{k}^{2}-(k-1) n+2 & \text { other } \end{array}\right.$$ then $G_{k,n}$ is hamiltonian. And the result may be the best

Tight cycles and regular slices in dense hypergraphs

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results. Firstly, we prove a hypergraph extension of the Erd\H{o}s-Gallai Theorem: for every δ>0 every sufficiently large k-uniform hypergraph with at least (α+δ)(nk) edges contains a tight cycle of length αn for each α∈[0,1]. Secondly, we find (asymptotically) the minimum codegree requirement for a k-uniform k-partite hypergraph, each of whose parts has n vertices, to contain a tight cycle of length αkn, for each 0<α<1

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum