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

Abstract

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 $u,v\geq m+2$.