Counting and packing Hamilton cycles in dense graphs and oriented graphs

TL;DR

This paper introduces a new method based on permanent estimates for counting and packing Hamilton cycles in dense graphs and oriented graphs, extending previous results and proving new extremal bounds.

Contribution

It develops a general approach for Hamilton cycle enumeration and packing, extending known results and proving new extremal theorems in dense graphs and oriented graphs.

Findings

Nearly $cn$-regular oriented graphs with $c>3/8$ contain exponentially many Hamilton cycles.

Graphs with minimum degree above half the vertices contain many edge-disjoint Hamilton cycles.

The results extend and generalize previous conjectures and theorems in Hamilton cycle enumeration.

Abstract

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ contains at least $(1-\varepsilon)\textrm{reg}_{even}(G)/2$ edge-disjoint Hamilton cycles, where $\reg(G)$ is the maximum \emph{even} degree of a spanning regular subgraph of $G$. This establishes an approximate version of a conjecture of K\"uhn, Lapinskas and Osthus.