4-cop-win graphs have at least 19 vertices
Combinatorics
2025-10-29 v2 Discrete Mathematics
Abstract
We show that the cop number of any graph on 18 or fewer vertices is at most 3. This answers a question posed by Andreae in 1986, as well as more recently by Baird et al. We also find all 3-cop-win graphs on 11 vertices, narrow down the possible 4-cop-win graphs on 19 vertices and make some progress on finding the minimum order of 3-cop-win planar graphs.
Keywords
Cite
@article{arxiv.2006.02998,
title = {4-cop-win graphs have at least 19 vertices},
author = {Jérémie Turcotte and Samuel Yvon},
journal= {arXiv preprint arXiv:2006.02998},
year = {2025}
}
Comments
31 pages, 9 figures. Data and code available at https://www.jeremieturcotte.com/research/min4cops/