Rainbow matchings of size $m$ in graphs with total color degree at least $2mn$

TL;DR

This paper generalizes existing results on rainbow matchings to graphs with a specified total color degree, establishing conditions under which such graphs contain large rainbow matchings.

Contribution

It introduces new conditions involving total color degree for the existence of rainbow matchings, extending prior work focused on minimum color degree or specific graph classes.

Findings

Graphs with total color degree 2mn contain rainbow matchings of size m under certain conditions.

Results apply to triangle-free, C4-free, properly colored, or sufficiently large graphs.

Generalizes previous theorems on rainbow matchings to broader classes of edge-colored graphs.

Abstract

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper, generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$; These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.