Characterization of forbidden subgraphs for bounded star chromatic number

Abstract

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are 2-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an H-avoidingk-coloring to be a proper k-coloring that forbids a bicolored subgraph H.When considering the class C of graphs with no F as an induced subgraph, it is not hard to see that every graph in C has bounded chromatic number if and only if F is a complete graph of size at most two. We study this phenomena for the class of graphs with no F as a subgraph for H-avoiding coloring. We completely characterize all graphs F where the class of graphs with no F as a subgraph has bounded H-avoiding chromatic number for a large class of graphs H. As a corollary, our main result implies a characterization of graphs F where the class of graphs with no F as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.