The number of Hamiltonian decompositions of regular graphs

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 Kühn 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 ( 1 + o ( 1 ) ) n 2 / 2 .