On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

TL;DR

This paper investigates the Hamilton-Waterloo problem for decompositions of complete graphs into 2-factors with cycles of different odd lengths, establishing existence results under specific divisibility and gcd conditions.

Contribution

It extends the Hamilton-Waterloo problem to cases with two cycle sizes of different parity, providing new existence results for such decompositions.

Findings

Existence of (2^kx,y)-HWP(vm;r,s) under gcd conditions

Decomposition results for odd cycle sizes with divisibility constraints

Exceptions when r or s equals 1

Abstract

The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair $(r,s)$. In this paper, we show that for odd $x$ and $y$, there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\gcd(x,y)\geq 3$, $m\geq 3$, and both $x$ and $y$ divide $v$, except possibly when $1\in\{r,s\}$.