Abelian groups without 3-chromatic Cayley graphs

TL;DR

This paper characterizes abelian groups that admit 3-chromatic Cayley graphs, showing it depends on the group's exponent, and connects graph chromatic properties with topological invariants of neighborhood complexes.

Contribution

It provides a complete characterization of abelian groups with 3-chromatic Cayley graphs and links topological properties of neighborhood complexes to graph coloring bounds.

Findings

Cayley graphs on abelian groups have chromatic number 3 iff the group’s exponent is not 1, 2, or 4.

For finitely generated groups, the characterization extends to connected Cayley graphs.

Topological invariants like torsion in homology imply chromatic number at least 4 for certain graphs.

Abstract

Let $G$ be an abelian group. The main theorem of this paper asserts that there exists a Cayley graph on $G$ with chromatic number $3$ if and only if $G$ is not of exponent $1$, $2$, or $4$. For connected Cayley graphs, we also show that this theorem holds when $G$ is finitely generated. Although motivated by ideas from algebraic topology, our proof may be expressed purely combinatorially. As a by-product, we derive a topological result which is of independent interest. Suppose $X$ is a connected non-bipartite graph, and let $\N(X)$ denote its neighborhood complex. We show that if the fundamental group $\pi_1(\N(X))$ or first homology group $H_1(\N(X))$ is torsion, then the chromatic number of $X$ is at least $4$. This strengthens a special case of a classical result of Lov\'asz, which derives the same conclusion if $\pi_1(\N(X))$ is trivial.