On the Hamilton-Waterloo Problem with a single factor of 6-cycles

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M,C_N)$-decomposition of $K_v$ into $\alpha$ $C_M$-factors and $\beta$ $C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by $\text{HWP}(v;M,N;\alpha ,\beta )$. Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the $\gcd (M,N)=\{2,3\}$, with a specific focus on the parameter $M=6$. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform $2$-factorizations. This innovative method not only extends the scope of solved cases, but also contributes to a deeper understanding of the complexity involved in solving the Hamilton-Waterloo Problem.