Topologically $4$-chromatic graphs and signatures of odd cycles

TL;DR

This paper explores the relationship between group-theoretic signatures of odd cycles in graphs and topological obstructions to 3-colorability, establishing conditions under which graphs are at least 4-chromatic.

Contribution

It introduces new connections between signatures from free groups and elementary abelian 2-groups and topological invariants related to graph colorability.

Findings

Existence of odd cycle with trivial free group signature implies high coindex of hom-complex.

Existence of odd cycle with trivial abelian 2-group signature implies high index of hom-complex.

Results provide new topological criteria for 4-chromatic graphs.

Abstract

We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which again implies that the chromatic number is at least 4).