In this paper, we prove that every 3-chromatic connected graph, except $C_7$ , admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path with 3 vertices. It is a special case of a conjecture due to S. Akbari, F. Khaghanpoor, and S. Moazzeni stating that every connected graph $G$ other than $C_7$ admits a $_X(G)$-coloring such that every vertex of $G$ is the beginning of a colorful path (i.e. a path on $_X(G)$ vertices containing a vertex of each color). We also provide some support for the conjecture in the case of 4-chromatic graphs.