On the Hamilton-Waterloo Problem with cycle lengths of distinct parities

Let $K_v^{\ast }$ denote the complete graph $K_v$ if $v$ is odd and $K_v-I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v,M,N,\alpha ,\beta$, the Hamilton-Waterloo problem asks for a $2$-factorization of $K_v^{\ast }$ into $\alpha$ $C_M$-factors and $\beta$ $C_N$-factors. Clearly, $M,N\ge 3$, $M\mid v$, $N\mid v$ and $\alpha+\beta = \lfloor\frac{v-1}{2}\rfloor$ are necessary conditions. Very little is known on the case where $M$ and $N$ have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever $M\mathrm{\mid }N$, $v>6N>36M$, and $\beta \ge 3$.