Akbari–Khaghanpoor–Moazzeni colorful-path conjecture

Let $G$ be a connected graph, let $\chi(G)$ denote its chromatic number, and call a path colorful under a proper $\chi(G)$-coloring when its $\chi(G)$ vertices have pairwise distinct colors. Akbari–Khaghanpoor–Moazzeni's conjecture. 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. This conjecture asks for a coloring that provides a colorful path starting at every vertex; the paper studies this conjecture and proves special cases, while the general assertion remains open.