The star and biclique coloring and choosability problems

TL;DR

This paper investigates the computational complexity of star and biclique coloring and choosability problems, establishing their hardness levels and exploring these problems in various specialized graph classes.

Contribution

It proves the complexity classes of star and biclique coloring and choosability problems are ^p-complete and ^p-complete respectively, even in restricted graph classes, and analyzes these problems in various graph subclasses.

Findings

Star and biclique k-coloring are ^p-complete for k > 2.

Star and biclique k-choosability are ^p-complete for k > 2.

Complexity results hold even in graphs without induced C_4 or K_{k+2}.

Abstract

A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic. If G admits a star (biclique) L-coloring for every k-list assignment L, then G is said to be star (biclique) k-choosable. In this article we study the computational complexity of the star and biclique coloring and choosability problems. Specifically, we prove that the star (biclique) k-coloring and k-choosability problems are \Sigma_2^p-complete and \Pi_3^p-complete for k > 2, respectively, even when the input graph contains no induced C_4 or K_{k+2}. Then, we study all these problems in some related classes of graphs, including H-free graphs for every H on three vertices, graphs with restricted diamonds, split graphs, threshold graphs, and net-free block graphs.