On the complexity of the colorful directed paths in vertex coloring of digraphs

The colorful paths and rainbow paths have been considered by several authors.A colorful directed path in a digraph $G$ is a directed path with $chi(G)$ vertices whose colors are different. A $v$-colorful directed path is such a directed path, starting from $v$. We prove that for a given $3$-regular triangle-free digraph $G$ determining whether there is a proper $chi(G)$ coloring of $G$ such that for every $v in V (G)$, there exists a $v$-colorful directed path is $ mathbf{NP} $ complete.