Approximate Hamilton decompositions of robustly expanding regular digraphs

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 K¨uhn, 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 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.