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Abelian groups without 3-chromatic Cayley graphs

Mike Krebs, Maya Sankar

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 $π_1(\N(X))$ 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 Lovász, which derives the same conclusion if $π_1(\N(X))$ is trivial.

Abelian groups without 3-chromatic Cayley graphs

Abstract

Let be an abelian group. The main theorem of this paper asserts that there exists a Cayley graph on with chromatic number if and only if is not of exponent , , or . For connected Cayley graphs, we also show that this theorem holds when 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 is a connected non-bipartite graph, and let denote its neighborhood complex. We show that if the fundamental group or first homology group is torsion, then the chromatic number of is at least . This strengthens a special case of a classical result of Lovász, which derives the same conclusion if is trivial.

Paper Structure

This paper contains 13 sections, 11 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Groups of Exponent Not Dividing $4$
  3. A discrete variant of the fundamental group
  4. The discrete fundamental group
  5. The discrete fundamental group of an abelian Cayley graph
  6. Winding number
  7. Obstructions to $3$-colorability
  8. Further Remarks
  9. Infinite groups and the axiom of choice
  10. Generalizing \ref{['thm:chi3iff']}
  11. Topological extensions
  12. Limitations of topological techniques
  13. Quadrangulations of surfaces

Key Result

Theorem 1.1

If $G=({\mathbb Z}/2{\mathbb Z})^m$ and $S$ is a symmetric subset of $G$, then $\chi(\mathop{\mathrm{Cay}}\nolimits(G,S))\neq 3$.

Figures (2)

  • Figure 1: A 3-coloring of the graph $\mathop{\mathrm{Cay}}\nolimits({\mathbb Z}/8{\mathbb Z},\{\pm 1,4\})$
  • Figure 2: A 3-coloring of the graph $\mathop{\mathrm{Cay}}\nolimits({\mathbb Z}/2^k{\mathbb Z},\{\pm 1,2^{m-1}\})$, which is isomorphic to $X_1$.

Theorems & Definitions (25)

  • Theorem 1.1: Payan Pay
  • Theorem 1.2
  • Theorem 1.3: Lovász Lov
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • ...and 15 more