It is known that the cop number c(G) of a connected graph G can be bounded as a function of the genus of the graph g(G). The best known bound, that c(G)≤⌊23g(G)⌋+3, was given by Schr\"{o}der, who conjectured that in fact c(G)≤g(G)+3. We give the first improvement to Schr\"{o}der's bound, showing that c(G)≤34g(G)+310.
@article{arxiv.1911.01758,
title = {Bounding the cop number of a graph by its genus},
author = {Nathan Bowler and Joshua Erde and Florian Lehner and Max Pitz},
journal= {arXiv preprint arXiv:1911.01758},
year = {2019}
}