Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

Abstract

AbstractWe show that for any fixed integer , a graph of maximum defiggree has a coloring with colors in which every connected bicolored subgraph contains at most edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which colors suffice, and acyclic colorings, for which colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree has a proper coloring with colors in which every bicolored subgraph is planar, as well as a proper coloring with colors in which every bicolored subgraph has treewidth at most 3.