On the Hamilton-Waterloo Problem with odd orders

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, $n,m\ge 3$, $m\mid v$,$n\mid v$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 $\beta >(n+5)\mathrm{/}2$ except possibly when $(m,\alpha) = (3,2)$,$(3,4)$,withsevenfurtherpossibleexceptionsin$(m,n,\alpha,\beta)$\mathrm{.}Wealsoshowthatwhen$n\geq m\geq 3$ 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 $0\le \alpha \le n$, $\beta =n-\alpha$, except possibly when $\alpha =2,4$, $\beta =1,3$, with three additional possible exceptions in $(m,n,\alpha )$.