Decomposing hypergraphs into cycle factors

Abstract

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum (k−1)-degree δk−1(H)⩾(1/2+o(1))n, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δ(G)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δk−1(H)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.