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arXiv · 1210.7269

The star and biclique coloring and choosability problems

Marina Groshaus·Francisco J. Soulignac·Pablo Terlisky
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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 Σ_2^p-complete and Π_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.

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Marina Groshaus, Francisco J. Soulignac, Pablo Terlisky. 2012-11-17. The star and biclique coloring and choosability problems. https://doi.org/10.7155/jgaa.00326

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