Chromatic numbers of rank-two Abelian Cayley graphs

TL;DR

This paper determines the chromatic number of rank-two Abelian Cayley graphs, completing the classification for all such graphs based on their Heuberger matrices, especially for higher dimensions.

Contribution

It provides a complete characterization of the chromatic number for all rank-two Abelian Cayley graphs, extending previous results to higher dimensions.

Findings

Chromatic number is 3 unless the graph has loops or is bipartite.

Chromatic number is 2 if the graph is bipartite.

Zero rows in the matrix do not affect the chromatic number.

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).