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

Abstract

We show that for any fixed integer ≥ m 1, a graph of maximum defiggree Δ has a coloring with ∕ O (Δ ) m m ( +1) colors in which every connected bicolored subgraph contains at most m 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 ∕ O (Δ ) 3 2 colors suffice, and acyclic colorings, for which ∕ O (Δ ) 4 3 colors suffice. Our proof uses a probabilistic