# A note on randomly colored matchings in random bipartite graphs

## TL;DR

This paper investigates the structure of perfect matchings in randomly colored bipartite graphs, providing bounds on the distribution of edge colors within such matchings.

## Contribution

It introduces the perfect matching color profile and derives bounds for this profile in the context of random bipartite graphs with random edge coloring.

## Key findings

- Bounds on the matching color profile for random bipartite graphs.
- Characterization of perfect matchings with specific color distributions.
- Insights into the interplay between randomness in graph structure and coloring.

## Abstract

We are given a bipartite graph that contains at least one perfect matching and where each edge is colored from a set $Q=\{c_1,c_2,\ldots,c_q}\$. Let $Q_i=\set{e\in E(G):c(e)=c_i}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_1,m_2,\ldots,m_q)\in [n]^q$ such that there exists a perfect matching $M$ such that $|M\cap Q_i|=m_i$. We give bounds on the matching color profile for a randomly colored random bipartite graph.