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arXiv · 2511.10813

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

Katherine Eng·Timothy Harris·Mike Krebs·Mason Meeks·Claudia Maria Schmidt
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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 $2$.

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Katherine Eng, Timothy Harris, Mike Krebs, Mason Meeks, Claudia Maria Schmidt. 2025-11-13. Four plane unit vectors generate a $3$-colorable graph. https://arxiv.org/abs/2511.10813

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