$m$-Cycle Packings of $(\lambda+\mu)K_{v+u}-\lambda K_v$: $m$ even

TL;DR

This paper provides a complete solution for decomposing a specific class of graphs into even cycles, extending previous work limited to small cycle lengths.

Contribution

It offers a comprehensive solution for decomposing $(rac{ ext{lambda}+ ext{mu}}{K_{v+u}}- ext{lambda} K_v)$ into even cycles for all even cycle lengths when $u,v geq m+2$.

Findings

Complete solution for even cycle decompositions

Applicable for all even cycle lengths with $u,v geq m+2$

Extends previous partial results for $m=3,5$

Abstract

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. 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 $u,v\geq m+2$.