On the Hamilton-Waterloo Problem with odd orders

Abstract

Given non-negative integers $v, m, n, \alpha, \beta$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. Clearly, $v$ odd, $n,m\geq 3$, $m\mid v$, $n\mid v$ and $\alpha+\beta = (v-1)/2$ are necessary conditions. To date results have only been found for specific values of $m$ and $n$. In this paper we show that for any $m$ and $n$ the necessary conditions are sufficient when $v$ is a multiple of $mn$ and $v>mn$, except possibly when $\beta=1$ or 3, with five additional possible exceptions in $(m,n,\beta)$. For the case where $v=mn$ we show sufficiency when $\beta > (n+5)/2$ except possibly when $(m,\alpha) = (3,2)$, $(3,4)$, with seven further possible exceptions in $(m,n,\alpha,\beta)$. We also show that when $n\geq m\geq 3$ are odd integers, the lexicographic product of $C_m$ with the empty graph of order $n$ has a factorization into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors for every $0\leq \alpha \leq n$, $\beta = n-\alpha$, except possibly when $\alpha= 2,4$, $\beta = 1, 3$, with three additional possible exceptions in $(m,n,\alpha)$.