We consider a variation of cop vs. robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on V (G) and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected n-vertex graph is exactly n. We also give bounds on the (generally greater) expected capture time when the gambler's distribution is unknown to the cop.
We show that the expected time for a smart "cop" to catch a drunk "robber" on an n-vertex graph is at most n + o(n). More precisely, let G be a simple, connected, undirected graph with distinguished points u and v among its n vertices. A cop begins at u and a robber at v; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on G; the cop sees all and moves as she wishes, with the object of "capturing" the robber-that is, occupying the same vertex-in least expected time. We show that the cop succeeds in expected time no more than n + o(n). Since there are graphs in which capture time is at least n − o(n), this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.
We find a formula for the number of directed 5-cycles in a tournament in terms of its edge scores and use the formula to find upper and lower bounds on the number of 5-cycles in any n-tournament. In particular, we show that the maximum number of 5-cycles is asymptotically equal to 3 4 n 5 , the expected number 5-cycles in a random tournament (p = 1 2 ), with equality (up to order of magnitude) for almost all tournaments.
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