A Note on Hamilton Decompositions of Even-Regular Multigraphs

Abstract

In this note, we prove that every even regular multigraph on $n$ vertices with multiplicity at most $r$ and minimum degree at least $rn/2 + o(n)$ has a Hamilton decomposition. This generalises a result of Vaughan who proved an asymptotic version of the multigraph $1$-factorisation conjecture. We derive our result by proving a more general result which states that dense regular multidigraphs that are robust outexpanders have a Hamilton decomposition. This in turn is derived from the corresponding result of Kühn and Osthus about simple digraphs.