Colouring Strong Products

TL;DR

This paper develops techniques to bound the chromatic number of strong products of graphs, linking graph coloring with topics in metric theory, probability, and information theory.

Contribution

It introduces general methods for bounding chromatic numbers of strong graph products involving graphs of bounded treewidth and explores their connections to other mathematical fields.

Findings

Bounded chromatic number for strong products of graphs with bounded treewidth.

Links between graph coloring and asymptotic dimension, percolation, and Shannon capacity.

Techniques applicable to fractional, clustered, and defective colorings.

Abstract

Recent results show that several important graph classes can be embedded as subgraphs of strong products of simpler graphs classes (paths, small cliques, or graphs of bounded treewidth). This paper develops general techniques to bound the chromatic number (and its popular variants, such as fractional, clustered, or defective chromatic number) of the strong product of general graphs with simpler graphs classes, such as paths, and more generally graphs of bounded treewidth. We also highlight important links between the study of (fractional) clustered colouring of strong products and other topics, such as asymptotic dimension in metric theory and topology, site percolation in probability theory, and the Shannon capacity in information theory.