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

Zazil Santizo Huerta; Melissa Keranen

paper

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

arXiv:2402.10081

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 $HWP (v; M, N; α, β)$ . 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 $g cd (M, N) = {2, 3}$ , 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.

37 pages, 13 figures