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

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 $2$.

20 pages