Decomposing hypergraphs into cycle factors

• Published in: arXiv.org
• Main Authors: Joos, Felix, Kühn, Marcus, Schülke, Bjarne
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 13.04.2021
• Subjects: Apexes; Asymptotic methods; Decomposition; Graph theory; Graphs; Integers; Questions
• ISSN: 2331-8422

A famous result by R\"odl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in \(k\)-uniform hypergraphs \(H\) on \(n\) vertices with minimum \((k-1)\)-degree \(\delta_{k-1}(H)\geq (1/2+o(1))n\), thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on \(n\) vertices with \(\delta(G)\geq (1/2+o(1))n\) 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 K\"uhn, Lapinskas, and Osthus. We extend this result to hypergraphs; every \(k\)-uniform hypergraph \(H\) on \(n\) vertices with \(\delta_{k-1}(H)\geq (1/2+o(1))n\) 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, K\"uhn, 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.