The star and biclique coloring and choosability problems

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) Lcoloring 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 Σp2-complete and Π p 3-complete for k > 2, respectively, even when the input graph contains no induced C4 or Kk+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, and threshold graphs. Submitted: November 2012 Reviewed: March 2014 Revised: March 2014 Accepted: May 2014 Final: May 2014 Published: June 2014 Article type: Regular paper Communicated by: K. Kawarabayashi Marina Groshaus was supported by PICT ANPCyT Grants 2010-1970 and 2010-1629, UBACyT Grant 20020100300042, and PIP CONICET Grant 11220100100310. Francisco Soulignac was supported by PICT ANPCyT Grant 2010-1970 and UBACyT Grant