Restricted Coloring Problems and Forbidden Induced Subgraphs

An acyclic coloring of a graph is a proper coloring such that any two color classes induce a forest. A star coloring of a graph is an acyclic coloring with the further restriction that the forest induced by any two color classes is a disjoint collection of stars. We consider the behavior of these problems when restricted to certain classes of graphs. In particular, we give characterizations of the classes of graphs for which two or more of these restricted coloring problems are equivalent, in that they share the same set of solutions. Surprisingly, our characterizations of these classes in terms of forbidden induced subgraphs equate them with classes that are well-studied in the literature. We extend this framework to encompass other restricted coloring problems, both known and new, and outline a method for obtaining results similar to those given here. We also explore the algorithmic implications of these results in terms of finding optimal acyclic and star colorings on certain classes of graphs. We show that optimal acyclic colorings of certain subclasses of even-hole-free graphs can be found in polynomial time, and that optimal acyclic and star colorings of trivially perfect graphs can be found in linear time.