The Hamilton-Waterloo Problem with even cycle lengths

TL;DR

This paper advances the understanding of the Hamilton-Waterloo Problem with even cycle lengths by developing a new construction method that nearly solves the existence question for such factorizations, except for a few specific cases.

Contribution

It introduces a novel construction technique to generate larger cycle factorizations from existing ones, significantly extending known solutions for the Hamilton-Waterloo Problem with even cycles.

Findings

Established solutions for HWP with odd ms and f5s under necessary conditions.

Almost complete resolution of the existence problem for even cycles, barring specific exceptions.

Provided new constructions that expand the range of known factorizations.

Abstract

The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 \in \{\alpha,\beta\}$ or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v \equiv 2 \pmod{4}$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP$(v;2m,2n;\alpha,\beta)$ for odd $\alpha$ and $\beta$ whenever the obvious necessary conditions hold, except possibly if $\beta=1$; $\beta=3$ and $\gcd(m,n)=1$; $\alpha=1$; or $v=2mn/\gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.