The threshold for perfect matching color profile in random coloring of random bipartite graphs

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 \cite{F} defined a notion of 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 $\mcp(G) = [n]^q$, answering a question posed by Frieze.