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The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (CM, CN) -decomposition of Kᵥ into CM-factors and CN-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by HWP (v; M, N;, ). 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.
Huerta et al. (2024) studied this question.