John Asplund — Page 2

arXiv2016

$m$-Cycle Packings of $(λ+μ)K_{v+u}-λK_v$: $m$ even

A $λK_v$ is a complete graph on $v$ vertices with $λ$ edges between each pair of the $v$ vertices. A $(λ+μ)K_{v+u}-λK_v$ is a $(λ+μ)K_{v+u}$ with the edge set of $λK_v$ removed. Decomposing a $(λ+μ)K_{v+u}-λK_v$ into edge-disjoint $m$-cycles has been studied by many people. To date, there is a complete solution for $m=4$ and partial results when $m=3$ or $m=5$. In this paper, we are able to solve this problem for all even cycle lengths as long as $u,v\geq m+2$.

math.CO↗

arXiv2015

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

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.

math.CO↗