A note on large rainbow matchings in edge-coloured graphs

TL;DR

This paper improves the bounds on the number of vertices needed to guarantee large rainbow matchings in edge-coloured graphs with a given minimum colour degree, reducing the requirement from quadratic to linear in k.

Contribution

It establishes that for k ≥ 4, a graph with at least 4k-4 vertices guarantees a rainbow matching of size k, improving previous quadratic bounds.

Findings

For k ≥ 4, n ≥ 4k-4 suffices for a rainbow matching of size k.

Reduces the vertex bound from (17/4)k^2 to linear in k.

Provides a tighter condition for the existence of large rainbow matchings.

Abstract

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on $n$ vertices with minimum colour degree at least $k$ contains a rainbow matching of size at least $k$, provided $n\geq (17/4)k^2$. In this paper, we show that $n\geq 4k-4$ is sufficient for $k \ge 4$.