The number of Hamiltonian decompositions of regular graphs

Abstract

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kuhn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r (1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K n is $${n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}$$ .