Role coloring bipartite graphs

A k-role coloring of a graph G is an assignment of colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n-role coloring.

While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k-role coloring is a notoriously hard problem for k greater than or equal to 2. In fact, it is known that k-Role coloring is NP-complete for k at least 2 on general graph class.

There has been extensive research on the complexity of k-role coloring on various hereditary graph classes. Furthering this direction of research, we show that
k-Role coloring is NP-complete on bipartite graphs for k at least 3 (while it is trivial for k=2). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3-Role coloring for this class of graphs. We further show that 2-Role coloring is NP-complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d=1.