Rainbow Matchings: existence and counting

TL;DR

This paper investigates the existence and enumeration of rainbow matchings in edge-colored bipartite graphs, providing asymptotic counts and probabilistic results that relate to Latin transversals in matrices.

Contribution

It offers new asymptotic enumeration formulas for rainbow matchings and demonstrates probabilistic existence results in random edge-colored bipartite graphs.

Findings

Asymptotic enumeration of rainbow matchings based on color frequency

High probability of rainbow matchings in random models with at least n colors

Almost all matrices with entries appearing at most n times have Latin transversals

Abstract

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of occurrences of a color. We also consider two natural models of random edge-colored K_{n,n} and show that, if the number of colors is at least n, then there is with high probability a random matching. This in particular shows that almost every square matrix of order n in which every entry appears at most n times has a Latin transversal.