Large rainbow matchings in edge-colored graphs with given average color degree

TL;DR

This paper establishes new bounds for the existence of large rainbow matchings in edge-colored graphs based on average color degree and vertex count, improving previous results and providing sharp bounds for complete graphs.

Contribution

It introduces improved bounds for rainbow matchings in edge-colored graphs using average color degree and vertex count, extending prior work by Kritschgau.

Findings

Every edge-colored graph with |V(G)| >= 4k-4 and average color degree >= 2k-1 contains a rainbow matching of size k.

Strongly edge-colored graphs with average degree >= 2k-1 contain rainbow matchings of size at least k.

The bounds are sharp for complete graphs.

Abstract

A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|>=12k^2+4k, or G is a properly edge-colored graph with |V(G)|>=8k. In this paper, we show that every edge-colored graph G with |V(G)|>=4k-4 and average color degree at least 2k-1 contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least 2k-1 contains a rainbow matching of size at least k. The bound is sharp for complete graphs.