The capture time of the hypercube

TL;DR

This paper determines that the capture time of an n-dimensional hypercube in the game of Cops and Robbers grows proportionally to n log n, using a new randomized strategy linked to coupon-collector analysis.

Contribution

It introduces a novel randomized strategy and proves the asymptotic capture time of hypercubes, advancing understanding of pursuit-evasion dynamics on high-dimensional graphs.

Findings

Capture time of hypercube is Θ(n log n).

New randomized strategy involving coupon-collector problem analysis.

Provides bounds on pursuit-evasion in high-dimensional graphs.

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.