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

TL;DR

The paper proves that any Cayley graph on a planar four-vector abelian system, generated by $\{\pm v_1,\pm v_2,\pm v_3,\pm v_4\}$ with planar unit vectors $v_i$, is 3-colorable. It develops a standardized abelian Cayley graph framework using Heuberger matrices and leverages modified Hermite Normal Form to reduce the problem to the $4\times2$ case, yielding a sharp dichotomy: $\chi(X)=3$ in general, and $\chi(X)=4$ only when there exist a signed permutation $P$ and unimodular $U$ with $PMU=1a1b1c01$ for integers $a,b,c$ with $3\mid (a+b+c)$. The authors determine the exact values $\chi_{\max}(1)=\chi_{\max}(2)=2$ and $\chi_{\max}(3)=\chi_{\max}(4)=3$, connecting to the Hadwiger–Nelson problem via de Bruijn–Erdős under AC. The work combines structural matrix methods, graph homomorphisms, and constructive colorings on triangular lattices to establish 3-colorability and lays groundwork for extending these techniques to larger generator sets.

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 $2$.

Theorems & Definitions (20)

• Corollary 1.1
• Theorem 1.2
• Proposition 2.1: Cervantes_2023
• Proposition 2.2: Cervantes_2023
• Proposition 2.3: Cervantes_2023
• Proposition 2.4: Tomato Cage Theorem, Cervantes_2023
• proof
• Proposition 4.1: Cervantes_2023
• Proposition 4.2: Cervantes_2023
• Proposition 4.3: Cervantes_2023