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

Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \set{c_1,c_2, \ldots, c_q}$\mathrm{.}Let$Q_i$bethesetofalledgescoloredwith$c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\text{mcp}(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q) \in [n]^q$suchthatthereexistsaperfectmatching$M$in$G$with$|Q_i \cap M| = m_i$forall$i$\mathrm{.}Let$\a_1, \a_2, \ldots, \a_q$ be positive constants such that ${\sum }_{i=1}^q{\text{a}}_i=1$. 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 $\text{mcp}(G)=\{(m_1,\dots ,m_q)\in [0,n{]}^q:m_1+\cdots +m_q=n\}$, 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 $G_{n,p}$.