Cops and robber in graphs with bounded vertex cover number

TL;DR

This paper extends bounds on the cop number in graphs by relating it to the vertex cover number, providing the first sublinear upper bound in terms of this parameter.

Contribution

It establishes a sublinear upper bound on the cop number for graphs with bounded vertex cover number, generalizing previous results related to Meyniel's conjecture.

Findings

Connected graphs with vertex cover number k have cop number at most k/2^{(1-o(1))\sqrt{\log k}}

First sublinear upper bound on cop number in terms of vertex cover number

Extends previous bounds related to Meyniel's conjecture

Abstract

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In this paper, we extend their result by showing that every connected graph with vertex cover number $k$ has cop number at most $k/2^{(1-o(1))\sqrt{\log k}}$. This is the first sublinear upper bound on the cop number in terms of the vertex cover number.