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

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (CM, CN)-decomposition of Kᵥ into α CM-factors and β CN-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by HWP(v; M, N; α, β). 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 (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.