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arXiv · 2402.10081

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

Zazil Santizo Huerta·Melissa Keranen
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Abstract

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 $\hbox{HWP}(v; M, N; \alpha, \beta)$. 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.

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Zazil Santizo Huerta, Melissa Keranen. 2024-02-15. On the Hamilton-Waterloo Problem with a single factor of 6-cycles. https://arxiv.org/abs/2402.10081

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