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

Abstract

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ 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 method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree $Δ$ has a proper coloring with $O(Δ^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(Δ^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.