Colorful paths for 3-chromatic graphs

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. It is a special case of a conjecture due to S.~Akbari, F.~Khaghanpoor, and S.~Moazzeni, cited in [P.J. Cameron, Research problems from the BCC22, \emph{Discrete Math.} {\bf 311} (2011), 1074--1083], stating that every connected graph $G$ other than $C_7$ admits a $\chi (G)$-coloring such that every vertex of $G$ is the beginning of a colorful path (i.e. a path of on $\chi(G)$ vertices containing a vertex of each color). We also provide some support for the conjecture in the case of 4-chromatic graphs.