Bounding the cop number of a graph by its genus

Nathan Bowler, Joshua Erde, Florian Lehner, Max Pitz

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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 $c(G) \leq \left\lfloor \frac{3 g(G)}{2}\right\rfloor + 3$, was given by Schr\"{o}der, who conjectured that in fact $c(G) \leq g(G) + 3$. We give the first improvement to Schr\"{o}der's bound, showing that $c(G) \leq \frac{4g(G)}{3} + \frac{10}{3}$.
Original languageEnglish
Pages (from-to)2459-2489
Number of pages31
JournalSIAM Journal on Discrete Mathematics
Issue number4
Publication statusPublished - 19 Nov 2021


  • cops and robbers
  • surfaces

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