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.
Highlights
Coloring problems are among the most studied problems in algorithmic graph theory
The star and biclique coloring and choosability problems are defined in Section 2, where we introduce the terminology that will be used throughout the article. for k ≥
We first conclude that the star 2-choosability problem is Πp3-complete with a polynomial-time reduction from the qsat3, and we show that star k-choosability can be reduced in polynomial time to the star (k + 1)-choosability
Summary
Coloring problems are among the most studied problems in algorithmic graph theory. In its classical form, the k-coloring problem asks if there is an assignment of k colors to the vertices of a graph in such a way that no edge is monochromatic. A graph G is clique (star, biclique) k-choosable when it admits an L-coloring generating no monochromatic maximal cliques (star, bicliques), for every k-list assignment L. In this paper we consider the star and biclique coloring and choosability problems, both for general graphs and for some restricted classes of graphs. In 2, Section 3, we prove and that it remains that the star k-coloring Σp2-complete even when problem its input is is. The remaining sections of the article study the star and biclique coloring problems on graphs with restricted inputs.
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