Bounding the cop number of a graph by its genus

TL;DR

This paper improves the upper bound on the cop number of a graph based on its genus, moving closer to Schr"oder's conjectured bound and advancing understanding in graph pursuit problems.

Contribution

The paper provides the first improvement to Schr"oder's bound, establishing a tighter upper bound of g(G)/3 + 10/3 for the cop number in terms of genus.

Findings

New upper bound: c(G) g(G)/3 + 10/3

Progress towards Schrf6der's conjecture

Enhanced understanding of cop number and genus relationship

Abstract

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 $c(G) \leq \frac{4g(G)}{3} + \frac{10}{3}$.