Existences of rainbow matchings and rainbow matching covers

TL;DR

This paper proves new bounds on the existence of large rainbow matchings in edge-coloured graphs based on minimum colour degree, and shows how to decompose graphs into rainbow matchings with optimal bounds.

Contribution

It improves previous bounds for rainbow matchings in graphs with given colour degree and maximum monochromatic degree, providing sharp results and new decomposition methods.

Findings

Graphs with minimum colour degree k contain rainbow matchings of size at least k for sufficiently large n.

Edge-decomposition of graphs into rainbow matchings is possible with bounds depending on maximum monochromatic degree.

Results are sharp and improve previous theorems by LeSaulnier and West.

Abstract

Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph such that its edges have distinct colours. The minimum colour degree $\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges incident with a vertex of $G$. We show that every edge-coloured graph $G$ on $n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching of size at least $k$, which improves the previous result for $k \ge 10$. Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and $\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most $\lfloor tn/2 \rfloor $ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West.