Plenary Talk

In this talk, we consider the pursuit-evasion game Cops and Robbers. The game is played on a graph between two players: a set of cops and a single robber, who take turns moving along the edges. The cop number of a graph is the minimum number of cops needed to guarantee capture of the robber, meaning they eventually occupy the same vertex. This parameter has been studied on a wide range of graph classes.

We discuss why studying roots of combinatorial polynomials plays an important role in understanding the underlying combinatorics of the structures these polynomials encode. We highlight the author's recent proof that real roots of all-terminal reliability polynomials of simple graphs are dense in U{1}; refining the same theorem proved for multigraphs by Brown and Colbourn (1992).

In this talk, we give an overview of the history of the Oberwolfach Problem and its directed variant. We then discuss a recent result which completely solves the directed Oberwolfach problem with bipartite 2-factors when the order n is congruent to 2 modulo 4. This is joint work with Peter Danziger and Alice Lacaze-Masmonteil.