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

the colorful paths and rainbow paths have been considered by severalauthors.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.