Decomposing hypergraphs into cycle factors

TL;DR

This paper extends a known result from graphs to hypergraphs, showing that hypergraphs with high minimum degree contain many edge-disjoint Hamilton cycles, and provides approximate decompositions into cycle factors.

Contribution

It generalizes the graph cycle decomposition result to hypergraphs, establishing the existence of multiple edge-disjoint Hamilton cycles under similar degree conditions.

Findings

Hypergraphs with minimum degree above (1/2+o(1))n contain (1-o(1))r edge-disjoint Hamilton cycles.

The result applies to approximately vertex-regular hypergraphs with quasirandom properties.

Provides approximate decompositions into cycle factors without short cycles.

Abstract

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi 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.