A note on rainbow matchings in properly edge-coloured graphs

TL;DR

This paper proves that properly edge-coloured graphs with a certain size condition always contain a rainbow matching of size equal to the minimum degree, improving previous bounds in the field.

Contribution

The authors establish a new size condition ensuring the existence of large rainbow matchings in properly edge-coloured graphs, refining earlier results.

Findings

Every properly edge-coloured graph with at least (9δ(G)-5)/2 vertices has a rainbow matching of size δ(G).

The result improves previous bounds on rainbow matchings in such graphs.

Provides a tighter relationship between graph size and rainbow matching size.

Abstract

A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9\delta(G) -5)/2$ has a rainbow matching of size $\delta(G)$, improving a result of Diemunsch et al.