PreprintCC BY 4.0arXiv
A planar signed graph with circular chromatic number five
Abstract
A circular r-coloring of a signed graph maps its vertices to a circle of circumference r≥2, with the endpoints of negative edges at circular distance at least $1$ and those of positive edges at most r/2−1; the circular chromatic number is the infimum of such r. We construct a simple planar signed triangulation on $85$ vertices with circular chromatic number exactly $5$. This improves the lower bound on the supremum of the circular chromatic numbers of simple planar signed graphs from 14/3 to $5$, giving a partial answer to a question posed by Naserasr, Wang, and Zhu.
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BibTeX
@misc{wang2026planar,
title = {{A planar signed graph with circular chromatic number five}},
author = {Lujia Wang and Huan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34118},
eprint = {2610.34118},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34118},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A planar signed graph with circular chromatic number five
AU - Lujia Wang
AU - Huan Zhou
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34118
UR - https://arxiv.org/abs/2610.34118
AB - A circular r-coloring of a signed graph maps its vertices to a circle of circumference r≥2, with the endpoints of negative edges at circular distance at least $1$ and those of positive edges at most r/2−1; the circular chromatic number is the infimum of such r. We construct a simple planar signed triangulation on $85$ vertices with circular chromatic number exactly $5$. This improves the lower bound on the supremum of the circular chromatic numbers of simple planar signed graphs from 14/3 to $5$, giving a partial answer to a question posed by Naserasr, Wang, and Zhu.
ER -