Abstract: We consider ``Containment'': a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by ``containing'' the robber---that is, by occupying all of the edges incident with a vertex v while the robber is at v. We develop several bounds on the minimal number of cops required to contain a robber, in particular relating this number to the well-studied ``cop-number'' in the original Cops and Robber game. We discuss the containability and containment number of some specific families of graphs, including hypercubes and certain Cartesian products. Time permitting, we will conclude with some discussion of current conjectures and other directions for future work.
Slides are available here.

Abstract: We will discuss a bit of the motivation for and history of the problem of computing the number of k-cycles in tournaments, as well as some new results. Specifically, we will see that while the maximum number of directed 3-cycles in a tournament is asymptotically equal to the expected number, this is not true for 4-cycles. The natural extensions (to the cases where k > 4) have largely remained open for decades. We will compute a formula for the number of 5-cycles in any tournament, and use it to show that the number of 5-cycles in a tournament cannot exceed the expected number. Time permitting, we will conclude with a summary of some results that are known for k > 5 and some open problems.
Slides are available here.

Abstract: We consider a pursuit-evasion game played on a graph in which the pursuer (whom we'll call the "hunter") is not constrained by the graph but must play in the dark against a "mole.'' On what kinds of graphs can the hunter guarantee capture of the mole in bounded time? It turns out that a graph is hunter-win if and only if it is a "lobster." We'll see exactly what lobsters are, define an optimal hunter strategy (and consequently an upper bound on maximum game time on hunter-win graphs) which surprisingly does not take advantage of the hunter's unconstrained movement, and discuss potential generalizations and open problems resulting from this work.
Slides are available here.

Abstract: We consider a variation of cop vs. robber on graph, representing a situation in which an anti-incursion is navigating a linked list of ports attempting to intercept an enemy packet in minimal expected time. The robber/enemy is not restricted by the graph edges and instead picks a time-independent probability distribution on V(G) and moves according to this fixed distribution. The cop/anti-incursion program moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that -- rather surprisingly! -- when the distribution is known, the expected capture time (with best play) on any connected n-vertex graph is exactly n. Time permitting, we will also discuss what bounds are known about the case of an unknown distribution.

Abstract: We examine variations of cops and robbers games on graphs. Our goals are to introduce some randomness into their study, and to estimate (expected) capture time. We show that a cop chasing a random walker can capture him in expected time n+o(n). We also discuss games in which the players move in the dark (showing that a cop can capture an immobile hider in time n on any graph) and in which the players suffer various restrictions on their movements.
Slides are available here.

Abstract: We consider a variation on the cops and robbers pursuit-evasion game in which the robber proceeds according to a random walk, and address the question of (expected) capture time. It turns out that the drunk can be captured in n + o(n) steps by a somewhat intelligent cop. We will motivate this result by showing examples of graphs that confound cops that follow a couple of obvious strategies. Time permitting, we will briefly discuss the intuition behind the proof of this result.