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=1 edges at a time, establishing a general upper bound of , where a = 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 n1 - 1/(R - 2) for finite R=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. (c) 2011 Wiley Periodicals, Inc. J Graph Theory.