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

TL;DR

This paper establishes new upper bounds on the number of colors needed for graph colorings with restrictions on bicolored subgraphs, unifying and extending previous results for acyclic, star, and treewidth colorings.

Contribution

It introduces a unified probabilistic approach to bound the number of colors for various restricted bicolored subgraph colorings, including new bounds for planarity and treewidth constraints.

Findings

Graphs with maximum degree Δ can be colored with O(Δ^{(m+1)/m}) colors for restricted bicolored subgraphs.

New bounds include O(Δ^{9/8}) colors for planar bicolored subgraphs.

Additional bounds include O(Δ^{13/12}) colors for subgraphs with treewidth at most 3.

Abstract

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ 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 $O(\Delta^{3/2})$ colors suffice, and acyclic colorings, for which $O(\Delta^{4/3})$ 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 $\Delta$ has a proper coloring with $O(\Delta^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(\Delta^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.