Role coloring of graphs from rooted products - Komathi, M. (Прикладная дискретная математика)
A k-role coloring is an assignment of k colors to the vertices of a graph such that if any two vertices receive the same color, then the set of colors assigned to their neighborhood will also be the same. Any graph with n vertices can have n-role coloring. Although it is easy to determine whether a graph with n vertices accepts a 1-role coloring, the challenge of k-role coloring is known to be difficult for k > 2. In fact, k-role coloring is known to be NP-complete for k > 2 on general graphs. In this paper, we determine k-role coloring of the rooted product of various graphs.