A Bound for the Cops and Robbers Problem

In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph G. Cops and a robber are allowed to move along the edges of G, and the goal of cops is to capture the robber. The cop number cðGÞ of G is the minimum number of cops required to win the game. Meyniel conjectured a long time ago thatOð ffiffiffi n p Þ cops are enough for any connectedG on n vertices. Improving several previous results, we prove that the cop number of an n-vertex graph is at most n2−ð1þoð1ÞÞ ffiffiffiffiffiffiffiffi log n p . A similar result independently and slightly before us was also obtained by Lu and Peng.