Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs

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” outneighborhood of $S$ is a little larger than $S$. This generalizes a result of Kühn, Osthus, and Treglown on approximate Hamilton decompositions of dense regular oriented graphs. It also generalizes a result of Frieze and Krivelevich on approximate Hamilton decompositions of quasirandom (di)graphs. In turn, our result is used as a tool by Kühn 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.