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

Document Type

Article

Publication Date

7-4-2025

Abstract

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-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 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.

Publication Title

Ars Mathematica Contemporanea

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