Four plane unit vectors generate a $3$-colorable graph
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908651210407936 |
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| author | Eng, Katherine Harris, Timothy Krebs, Mike Meeks, Mason Schmidt, Claudia Maria |
| author_facet | Eng, Katherine Harris, Timothy Krebs, Mike Meeks, Mason Schmidt, Claudia Maria |
| contents | 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Four plane unit vectors generate a $3$-colorable graph Eng, Katherine Harris, Timothy Krebs, Mike Meeks, Mason Schmidt, Claudia Maria Combinatorics 05C25 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 $2$. |
| title | Four plane unit vectors generate a $3$-colorable graph |
| topic | Combinatorics 05C25 |
| url | https://arxiv.org/abs/2511.10813 |