Approximate Hamilton decompositions of robustly expanding regular digraphs

TL;DR

This paper proves that large, regular, robustly expanding directed graphs can be approximately decomposed into nearly edge-disjoint Hamilton cycles, extending previous results to broader classes of graphs.

Contribution

It generalizes existing theorems by showing approximate Hamilton decompositions in robust outexpanders, paving the way for exact decompositions.

Findings

Large regular robust outexpanders contain r-o(r) edge-disjoint Hamilton cycles.

The results extend to dense regular oriented graphs and quasirandom directed graphs.

The findings enable proofs of Hamilton decompositions in these graph classes.

Abstract

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the `robust' outneighbourhood of S is a little larger than S. This generalises a result of K\"uhn, Osthus and Treglown on approximate Hamilton decompositions of dense regular oriented graphs. It also generalises a result of Frieze and Krivelevich on approximate Hamilton decompositions of quasirandom (di)graphs. In turn, our result is used as a tool by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G which has linear degree and is a robust outexpander even has a Hamilton decomposition.