What is the name of the problem? (partitioning graph into three covers)
Your problem can be solved in linear time, by reduction to 2SAT. For each vertex $v$ we will have three variables $v_R,v_B,v_G$ and clauses $\lnot v_R \lor \lnot v_B,\lnot v_R \lor \lnot v_G, \lnot v_B \lor \lnot v_G$. These ensure that at most one of $v_R,v_B,v_G$ is true. For each edge $(v,w)$ labeled $R$, we will add the clause $v_R \lor w_R$. If there is a valid vertex coloring in your sense, then it clearly translates to a solution of this 2SAT instance. Conversely, any solution to the 2SAT instance corresponds to a partial coloring in which each edge is incident to a vertex with the same color. Coloring the other vertices arbitrarily, we obtain a valid vertex coloring in your sense.