Abstract
A famous result by Rödl, RuciĆski, and SzemerĂ©di guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum (kâ1)-degree ÎŽkâ1(H)â©Ÿ(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 ÎŽ(G)â©Ÿ(1/2+o(1))n contains (1âo(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 KĂŒhn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with ÎŽkâ1(H)â©Ÿ(1/2+o(1))n contains (1âo(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, KĂŒhn, 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.
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