Counting Hamilton decompositions of oriented graphs

A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus [15], who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r = cn for some fixed c > 3/8 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating the number of such decompositions of G and show that this number is n(1−o(1))cn 2 . In addition, we also obtain a new and much simpler proof for the approximate version of Kelly’s conjecture.