Variations on cops and robbers

Abstract

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of n/α√ log α n , where α = 1 + 1 R , thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an n-vertex graph can be as large as n 1− 1 R−2 for finite R, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is at most O(n(log log n) 2 / log n). Our approach is based on expansion.

Chasing a Fast Robber on Planar Graphs and Random Graphs

“…An intriguing fact about the Cops and Robber game is that although many scholars have studied the game it is not yet well understood. In particular, although the upper bound was conjectured already in 1987, no upper bound better than has been proved since then (see ).…”

Cops and Robbers playing on edges

“…√ log 2 n (which is still n 1−o(1) ) for any connected graph on n vertices (the result obtained independently by Lu and Peng [21], Scott and Sudakov [30], and Frieze, Krivelevich, and Loh [15]). …”

Meyniel's conjecture holds for random graphs

“…It took 20 years to show that c(n) = O(n/ log n) as proved in [7]. Today we know that the cop number is at most n2 −(1+o(1)) √ log 2 n (which is still n 1−o(1) ) for any connected graph on n vertices (the result obtained independently by Lu and Peng [14], Scott and Sudakov [20], and Frieze, Krivelevich and Loh [10]). If one looks for counterexamples for Meyniel's conjecture it is natural to study first the cop number of random graphs.…”