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Chromatic numbers of rank-two Abelian Cayley graphs

Mike Krebs, Alejandro Leyva

TL;DR

This work resolves the chromatic-number determination for rank-two Abelian Cayley graphs across all dimensions. By representing graphs via Heuberger matrices $M_X$ and exploiting graph-homomorphism techniques that merge rows, the authors establish that for $m\times 2$ matrices with $m\ge 5$ (no zero rows), a connected, loop-free, non-bipartite SACG has $\chi(X)=3$, while loops and bipartiteness give uncolorability and $\chi=2$, respectively; zero rows can be deleted without changing $\chi$. The proof hinges on a tight synthesis of prior rank-2 results (2×2, 3×2, 4×2) with a robust base case at $m=5$ and an induction step using $m\times 2$ to $(m-1)\times 2$ homomorphisms. The paper also posits a broader conjecture that, for fixed rank $r$, sufficiently large $m$ yields $\chi(X)=3$ for non-bipartite, loop-free, non-zero-row graphs, which holds for $r=1,2$. These results provide a concrete, algebraic handle on chromatic properties of Abelian Cayley graphs and suggest a path toward higher-rank generalizations.

Abstract

A connected Cayley graph for an Abelian group generated by a finite symmetric subset $S$ can be represented by an integer matrix, its Heuberger matrix. We call the number of columns of that matrix its rank and the number of rows its dimension. Several previous papers have dealt with the question of finding a formula for the chromatic number of an Abelian Cayley graph in terms of an associated Heuberger matrix. In this paper, we fully resolve this matter for all integer matrices of rank $\leq 2$. Prior results provide such formulas when the rank is 1, as well as when the rank is 2 and the dimension is no more than 4. Here, we complete the picture for the rank-two case by showing that when the rank is 2 and the dimension is at least 5, then the chromatic number equals 3 unless the graph has loops (in which case it is uncolorable); the graph is bipartite (in which case the chromatic number is 2); or the matrix has a zero row (in which case, the chromatic number does not change when that row is deleted).

Chromatic numbers of rank-two Abelian Cayley graphs

TL;DR

This work resolves the chromatic-number determination for rank-two Abelian Cayley graphs across all dimensions. By representing graphs via Heuberger matrices and exploiting graph-homomorphism techniques that merge rows, the authors establish that for matrices with (no zero rows), a connected, loop-free, non-bipartite SACG has , while loops and bipartiteness give uncolorability and , respectively; zero rows can be deleted without changing . The proof hinges on a tight synthesis of prior rank-2 results (2×2, 3×2, 4×2) with a robust base case at and an induction step using to homomorphisms. The paper also posits a broader conjecture that, for fixed rank , sufficiently large yields for non-bipartite, loop-free, non-zero-row graphs, which holds for . These results provide a concrete, algebraic handle on chromatic properties of Abelian Cayley graphs and suggest a path toward higher-rank generalizations.

Abstract

A connected Cayley graph for an Abelian group generated by a finite symmetric subset can be represented by an integer matrix, its Heuberger matrix. We call the number of columns of that matrix its rank and the number of rows its dimension. Several previous papers have dealt with the question of finding a formula for the chromatic number of an Abelian Cayley graph in terms of an associated Heuberger matrix. In this paper, we fully resolve this matter for all integer matrices of rank . Prior results provide such formulas when the rank is 1, as well as when the rank is 2 and the dimension is no more than 4. Here, we complete the picture for the rank-two case by showing that when the rank is 2 and the dimension is at least 5, then the chromatic number equals 3 unless the graph has loops (in which case it is uncolorable); the graph is bipartite (in which case the chromatic number is 2); or the matrix has a zero row (in which case, the chromatic number does not change when that row is deleted).

Paper Structure

This paper contains 13 sections, 18 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The past
  4. The present
  5. The future
  6. Previous Results
  7. $1\times 2$ Heuberger matrices
  8. $2\times 2$ Heuberger matrices
  9. $3\times 2$ Heuberger matrices
  10. $4\times 2$ Heuberger matrices
  11. Proof of \ref{['thm:main']}
  12. $5\times 2$ Heuberger matrices
  13. $m\times 2$ Heuberger matrices with $m>5$

Key Result

Theorem 1.1

Let $m\geq 5$ be an integer. Let $M_X$ be an $m\times 2$ integer matrix and $X=M_X^{SACG}$ (the Abelian Cayley graph associated with $M_X$). Suppose $M_X$ has no zero rows. If $X$ is not bipartite and has no loops, then $\chi(X)=3$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Conjecture 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Theorem 1.9: Tomato Cage Theorem
  • Lemma 2.1
  • ...and 14 more