Role Coloring Bipartite Graphs

TL;DR

This paper investigates the computational complexity of k-role coloring in bipartite graphs, establishing NP-completeness for k > 2 and providing polynomial algorithms for specific subclasses.

Contribution

It proves NP-completeness of k-role coloring for bipartite graphs when k > 2 and characterizes 3-role colorability in bipartite chain graphs, offering new complexity insights.

Findings

NP-completeness of k-role coloring for bipartite graphs when k > 2

Polynomial-time algorithm for 3-role coloring bipartite chain graphs

NP-completeness of 2-role coloring for graphs close to bipartite graphs

Abstract

A k-role coloring of a graph G is an assignment of k colors to the vertices of G such that 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 1. In fact, it is known that k-Role coloring is NP-complete for k greater than 1 on arbitrary graphs. 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 greater than 2 (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.