Rainbow Matchings of size \delta(G) in Properly Edge-colored Graphs

TL;DR

This paper proves that properly edge-colored graphs with minimum degree elta and sufficiently many vertices always contain a rainbow matching of size elta, and provides an efficient algorithm to find it.

Contribution

It confirms Wang's conjecture by establishing a linear bound on the number of vertices needed and offers a polynomial-time algorithm for constructing the rainbow matching.

Findings

Rainbow matching of size elta exists under given conditions

Bound of 6.5elta vertices suffices for the existence

Provides an elta|V(G)|^2-time algorithm for finding the matching

Abstract

A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(\delta) such that a properly edge-colored graph G with minimum degree \delta and order at least f(\delta) must have a rainbow matching of size \delta. We answer this question in the affirmative; f(\delta) = 6.5\delta suffices. Furthermore, the proof provides a O(\delta(G)|V(G)|^2)-time algorithm that generates such a matching.