The capture time of the hypercube

Abstract

In the game of Cops and Robbers, the capture time of a graph is the minimum number of moves needed by the cops to capture the robber, assuming optimal play. We prove that the capture time of the $n$-dimensional hypercube is $\Theta (n\ln n).$ Our methods include a novel randomized strategy for the players, which involves the analysis of the coupon-collector problem.