Tight Cycles in Hypergraphs

Abstract

Abstract. 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 Erdős-Gallai Theorem for graphs: For every δ > 0, every sufficiently large k-uniform hypergraph on n vertices with at least (α+δ) n k edges contains a tight cycle of length αn for any α ∈ [0, 1].Our second result concerns k-partite k-uniform hypergraphs with partition classes of size n and for each α ∈ (0, 1) provides an asymptotically optimal minimum codegree requirement for the hypergraph to contain a cycle of length αkn.