The Hamilton-Waterloo Problem with $C_4$ and $C_m$ Factors

Abstract

The Hamilton-Waterloo problem with uniform cycle sizes asks for a $2-$ factorization of the complete graph $K_v$ (for odd {\em v}) or $K_v$ minus a $1-$factor (for even {\em v}) where $r$ of the factors consist of $n-$cycles and $s$ of the factors consist of $m-$cycles with $r+s=\left \lfloor \frac{v-1}{2} \right \rfloor$. In this paper, the Hamilton-Waterloo Problem with $4-$cycle and $m-$cycle factors for odd $m\geq 3$ is studied and all possible solutions with a few possible exceptions are determined.