Colorful paths for 3-chromatic graphs

In this paper, we prove that every 3-chromatic connected graph, except C7, 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, Discrete Math. 311 (2011), 1074–1083], stating that every connected graph G other than C7 admits a χ(G)-coloring such that every vertex of G is the beginning of a colorful path (i.e. a path of on χ(G) vertices containing a vertex of each color). We also provide some support for the conjecture in the case of 4-chromatic graphs.