Large rainbow matchings in general graphs

TL;DR

This paper explores conditions for large rainbow matchings in general graphs, proposing conjectures based on bipartite graph results and proving a new bound for the existence of such matchings.

Contribution

It introduces conjectures extending bipartite graph results to general graphs and proves a new bound of 3n-2 matchings for rainbow matchings.

Findings

Proposes conjectures for rainbow matchings in general graphs based on parity of n.

Proves that 3n-2 matchings suffice for a rainbow matching of size n.

Extends known bipartite results to more general graph classes.

Abstract

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Bar\'at, Gy\'arf\'as and S\'ark\"ozy, we conjecture that if $n$ is odd then the same is true also in general graphs, and that if $n$ is even then $2n$ matchings of size $n$ suffice. We prove that any $3n-2$ matchings of size $n$ have a partial rainbow matching of size $n$.