Optimal packings of Hamilton cycles in graphs of high minimum degree

TL;DR

This paper determines the maximum number of edge-disjoint Hamilton cycles in large graphs with high minimum degree, extending known results and using advanced decomposition theorems.

Contribution

It provides an optimal characterization of the number of edge-disjoint Hamilton cycles in graphs with high minimum degree, generalizing previous results and applying recent Hamilton decomposition techniques.

Findings

Number of edge-disjoint Hamilton cycles equals reg_even(n,d)/2.

Extended results to graphs with minimum degree >= n/2.

Applied recent Hamilton decomposition theorems to robustly expanding graphs.

Abstract

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let reg_even(n,d) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree d. Then the number of edge-disjoint Hamilton cycles we find equals reg_even(n,d)/2. The value of reg_even(n,d) is known for infinitely many values of n and d. We also extend our results to graphs G of minimum degree d >= n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of K\"uhn and Osthus on Hamilton decomposition of robustly expanding regular graphs.