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

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 $\hbox{HWP}(v; M, N; α, β)$. 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.

Figures (11)

• Figure 1: Edge induced subgraph with cross differences $(0,1,1,2)$ on the 4-cycle $(0,2,1,3)$, and cross differences $(1,1,1)$ on the 3-cycle $(4,5,6)$
• Figure 2: Representation of ${d_k}^+$ and ${d_k}^-$.
• Figure 3: 1-factorization of $K_6$ into 5 1-factors.
• Figure 4: One 8-cycle factor in $K_{(6:4)}$ from a $MRSM_{\mathbb Z_6}(S,t,4;\Sigma)$ with row $\alpha = [3^+,3^-,3^+,3^-]$. The dashed edges represent the 4-cycles ${c'_0}, c'_2, c'_4$.
• Figure 5: A $2x$-cycle factor formed from cross differences zero and inside edges from $f_5$.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Corollary 6
• Theorem 7
• Lemma 8
• Corollary 9
• Lemma 10