The number of Hamiltonian decompositions of regular graphs

TL;DR

This paper estimates the number of Hamiltonian decompositions in regular graphs, extending classical results and providing an asymptotic count for such decompositions in complete graphs.

Contribution

It provides an asymptotic estimate for the number of Hamiltonian decompositions in regular graphs, generalizing previous existence results.

Findings

Number of Hamiltonian decompositions in regular graphs is approximately r^{(1+o(1))nr/2}

Number of decompositions in complete graphs is n^{(1-o(1))n^2/2}

Extends classical theorems to count decompositions asymptotically

Abstract

A Hamilton cycle in a graph $\Gamma$ is a cycle passing through every vertex of $\Gamma$. A Hamiltonian decomposition of $\Gamma$ 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 K\"{u}hn and Osthus. They proved that every $r$-regular $n$-vertex graph $\Gamma$ 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 $H(\Gamma)$, the number of such decompositions of $\Gamma$. Our main result is that $H(\Gamma)=r^{(1+o(1))nr/2}$. In particular, the number of Hamiltonian decompositions of $K_n$ is $n^{(1-o(1))n^2/2}$.