On the Hamilton-Waterloo Problem with triangle factors and $C_{3x}$-factors

Abstract

The Hamilton-Waterloo Problem (HWP) in the case of $C_m$-factors and $C_n$-factors asks if $K_v$, where $v$ is odd (or $K_v-F$, where $F$ is a 1-factor and $v$ is even), can be decomposed into r copies of a 2-factor made either entirely of $m$-cycles and $s$ copies of a 2-factor made entirely of $n$-cycles. In this paper, we give some general constructions for such decompositions and apply them to the case where $m=3$ and $n=3x$. We settle the problem for odd $v$, except for a finite number of $x$ values. When $v$ is even, we make significant progress on the problem, although open cases are left. In particular, the difficult case of $v$ even and $s=1$ is left open for many situations.

Cited by in corpus (2)

• The Hamilton-Waterloo problem with C8-factors and Cm-factors
• On the Hamilton-Waterloo Problem with odd orders