Cops and robber in graphs with bounded vertex cover number

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.