Variations on cops and robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≤yen;1 edges at a time, establishing a general upper bound of n/,α^(1-0(1)) √logαn, where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. 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≤yen;5, 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 O(n(loglogn)²/logn). Our approach is based on expansion.