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

2024/02/15 by Zazil Santizo Huerta, Huerta, Zazil Santizo, Melissa Keranen +1
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2402.10081

openalex publication_date 2024/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (C_M, C_N)-decomposition of K_v into α C_M-factors and β C_N-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by \hboxHWP(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.

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