Hamiltonian cycles and 1-factors in 5-regular graphs

TL;DR

This paper explores the existence and properties of 1-factorisations and Hamiltonian cycles in 5-regular graphs across various genera, providing new constructions, answering longstanding questions, and analyzing edge-Kempe classes.

Contribution

It proves the existence of infinitely many 5-regular graphs with specific 1-factorisation properties, including solutions to a problem posed in 1964, and introduces techniques for constructing graphs with high cyclic edge-connectivity.

Findings

Existence of infinitely many 5-regular graphs with prescribed numbers of perfect pairs.

Planar 5-connected 5-regular graphs can have zero perfect pairs in all 1-factorisations.

Certain conditions guarantee a linear number of 1-factorisations with at least one perfect pair.

Abstract

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g = 0$, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other hand, by the Four Colour Theorem and a result of Brinkmann and the first author, every planar 4-connected 5-regular graph satisfying a condition on its hamiltonian cycles has a linear number of 1-factorisations each containing at least one perfect pair. We also prove that every planar 5-connected 5-regular graph satisfying a stronger condition contains a 1-factorisation with at most nine perfect pairs, whence, every such graph admitting a 1-factorisation with ten perfect pairs has at least two edge-Kempe equivalence classes. The paper concludes with further results on edge-Kempe equivalence classes in planar 5-regular graphs.