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PreprintCC BY 4.0arXiv

A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane

  • Jan Kristian Haugland
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Abstract

With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.

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Cite

BibTeXRIS
BibTeX
@misc{haugland2026moserspindlefree,
  title = {{A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane}},
  author = {Jan Kristian Haugland},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.04542},
  eprint = {2608.04542},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.04542},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane
AU  - Jan Kristian Haugland
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.04542
UR  - https://arxiv.org/abs/2608.04542
AB  - With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.
ER  -