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On the Hamilton-Waterloo Problem with odd orders

Given non-negative integers $v, m, n, α, β$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $α$ $C_m$-factors and $β$ $C_n$-factors. Clearly, $v$ odd, $n,m\geq 3$, $m\mid v$, $n\mid v$ and $α+β= (v-1)/2$ 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 $β=1$ or 3, with five additional possible exceptions in $(m,n,β)$. For the case where $v=mn$ we show sufficiency when $β> (n+5)/2$ except possibly when $(m,α) = (3,2)$, $(3,4)$, with seven further possible exceptions in $(m,n,α,β)$. 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 $α$ $C_m$-factors and $β$ $C_n$-factors for every $0\leq α\leq n$, $β= n-α$, except possibly when $α= 2,4$, $β= 1, 3$, with three additional possible exceptions in $(m,n,α)$.