On the complexity of path problems in properly colored directed graphs

We address the complexity class of several problems related to finding a path in a properly colored directed graph. A properly colored graph is defined as a graph G whose vertex set is partitioned into $𝒳(G)$ stable subsets, where $𝒳(G)$ denotes the chromatic number of G. We show that to find a simple path that meets all the colors in a properly colored directed graph is NP‐complete, and so are the problems of finding a shortest and longest of such paths between two specific nodes.