Four plane unit vectors generate a $3$-colorable graph

TL;DR

This paper proves that any Cayley graph generated by four plane unit vectors and their negatives is always 3-colorable, extending to a broader class of abelian Cayley graphs with specific relation constraints.

Contribution

It establishes a universal 3-colorability result for Cayley graphs generated by four vectors under certain algebraic conditions, generalizing previous understanding.

Findings

Any such Cayley graph is 3-colorable.

The result applies to a broader class of abelian Cayley graphs with relation rank ≤ 2.

Provides a new link between geometric vectors and graph coloring.

Abstract

We show that given an arbitrary set of four plane unit vectors $v_1, v_2, v_3, v_4$, the Cayley graph generated by $\{\pm v_1, \pm v_2, \pm v_3, \pm v_4\}$ is always $3$-colorable. Indeed, we show that this is a specific case of a much more general result wherein we determine the chromatic number of an arbitrary abelian Cayley graph generated by a set of four elements and their negatives, subject to the constraint that the group of relations between those elements has rank no more than $2$.