Fractional cycle decompositions in hypergraphs

TL;DR

This paper proves that dense k-uniform hypergraphs can be fractionally decomposed into large cycles, providing near-complete cycle decompositions and solving a related problem for graphs using a new Markov chain method.

Contribution

It introduces a new method leveraging rapidly mixing Markov chains to find fractional cycle decompositions in hypergraphs and solves an open problem for graphs.

Findings

Fractional decompositions exist for dense hypergraphs into large cycles.

Approximate integral decompositions are achievable for these hypergraphs.

The method applies to graphs, guaranteeing integral cycle decompositions and resolving a known problem.

Abstract

We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into tight cycles of length $\ell$ ($\ell$-cycles for short) whenever $\ell\geq \ell_0$ and $n$ is large in terms of $\ell$. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into $\ell$-cycles. Moreover, for graphs this even guarantees integral decompositions into $\ell$-cycles and solves a problem posed by Glock, K\"uhn and Osthus. For our proof, we introduce a new method for finding a set of $\ell$-cycles such that every edge is contained in roughly the same number of $\ell$-cycles from this set by exploiting that certain Markov chains are rapidly mixing.