The threshold for the full perfect matching color profile in a random coloring of random graphs

TL;DR

This paper determines the threshold probability for the existence of a perfect matching with a specified color profile in randomly edge-colored bipartite and general random graphs, answering a question posed by Frieze.

Contribution

It establishes the precise threshold for the perfect matching color profile in random bipartite graphs and extends the results to general random graphs, advancing understanding of colored matchings.

Findings

Threshold for perfect matching color profile in bipartite graphs identified

Extended threshold results to general random graphs G_{n,p}

Answered a previously posed open question by Frieze

Abstract

Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \set{c_1,c_2, \ldots, c_q}$. Let $Q_i$ be the set of all edges colored with $c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\mcp(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q) \in [n]^q$ such that there exists a perfect matching $M$ in $G$ with $|Q_i \cap M| = m_i$ for all $i$. Let $\a_1, \a_2, \ldots, \a_q$ be positive constants such that $\sum_{i=1}^q \a_i = 1$. Let $G$ be the random bipartite graph $G_{n,n,p}$. Suppose the edges of $G$ are independently colored with color $c_i$ with probability $\alpha_i$. We determine the threshold for the event $\mcp(G) = \set{(m_1, \ldots, m_q) \in [0,n]^q : m_1 + \cdots + m_q = n}$, answering a question posed by Frieze. We further extend our methods to find the threshold for the same event in a randomly colored random graph $G_{n,p}$.