Cops and robbers in graphs with large girth and Cayley graphs

In [1] Aigner and Fromme and in [10] Quilliot studied the following game, called cops and robbers. There is a finite, connected, undirected graph G = (V, E), m cops and one robber. First the cops choose one vertex each as initial position. Next the robber makes his choice. Afterwards they move alternately (first the cops, then the robber) along the edges of the graph or stay. Denote by c(G) the minimum value of m for which m cops have a winning strategy, i.e., they have an algorithm to catch the robber (get on the same vertex as he) no matter how he plays. In [1] it is shown that c(G) is at least the minimum degree in graphs with girth 5 or more. Andreae [2] showed for every d > 3 the existence of regular graphs G of degree d and c(G) arbitrarily large-solving a problem of [1]. The main result of this paper extends the Theorem of Andreae.