S. Saqaeeyan - Abadan Branch, Islamic Azad University
Esmaeil Mollaahmadi - Sharif University of Technology .
Ali Dehghan - Amirkabir University of Technology, Tehran, Iran

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 ۳-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.

Colorful Directed Paths, Computational Complexity, Vertex Coloring