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

John Asplund

Published 2015-11-30Version 1

A $\lambda K_v$ is a complete graph on $v$ vertices with $\lambda$ edges between each pair of the $v$ vertices. A $(\lambda+\mu)K_{v+u}-\lambda K_v$ is a $(\lambda+\mu)K_{v+u}$ with the edge set of $\lambda K_v$ removed. Decomposing a $(\lambda+\mu)K_{v+u}-\lambda K_v$ into edge-disjoint $m$-cycles has been studied by many people within the last $10$ years. Past results have only managed to solve this problem 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 $u,v\geq m+2$.