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.

翻译：Meyniel猜想指出，$n$个顶点的连通图具有$O(\sqrt{n})$的警察数。目前已知的最佳上界是$n/2^{(1-o(1))\sqrt{\log n}}$，该结果由Lu和Peng（2011年）以及Scott和Sudakov（2011年）分别独立证明。在本文中，我们通过证明每个顶点覆盖数为$k$的连通图最多具有$k/2^{(1-o(1))\sqrt{\log k}}$的警察数，推广了他们的结果。这是首个关于警察数相对于顶点覆盖数的亚线性上界。