Rainbow matchings in properly-coloured multigraphs

TL;DR

This paper extends a conjecture about rainbow matchings from bipartite to general multigraphs, showing near-perfect rainbow matchings exist under certain edge multiplicity and colour conditions.

Contribution

It generalizes the rainbow matching conjecture to non-bipartite multigraphs with low edge multiplicities, establishing near-perfect rainbow matchings.

Findings

Existence of large rainbow matchings in multigraphs with low edge multiplicity

Conditions under which rainbow matchings of size close to n are guaranteed

Extension of bipartite rainbow matching results to general multigraphs

Abstract

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once.