Georgia Institute of Technology College of Sciences

Many results in asymptotic extremal combinatorics are obtained using just a handful of instruments, such as induction and Cauchy-Schwarz inequality. The ingenuity lies in combining these tools in just the right way. Recently, Razborov developed a flag calculus which captures many of the available techniques in pure form, and allows one, in particular, to computerize the search for the right combination. In this talk we outline the general approach and describe its application to the conjecture that a digraph with minimum outdegree n/3 contains a directed triangle. This special case of the Caccetta-Haggkvist conjecture has been extensively investigated in the past. We show that a digraph with minimum outdegree a*n contains a directed triangle for a = 0.3465. The proof builds on arguments used to establish previously known bounds, due to Shen from 1998 (a = 0.3542) and Hamburger, Haxell and Kostochka from 2008 (a = 0.3531). It consists of a combination of ~80 computer generated inequalities. Based on joint work with Jan Hladky and Daniel Kral.

Each graph can be embedded in 3-space. The problem becomes more interesting if we put restrictions on the type of embedding. For example, a linkless embedding of a graph is one where each pair of vertex-disjoint circuits has linking number equal to zero. The class of all graphs that have a linkless embedding is closed under taking minors. Robertson, Seymour, and Thomas gave the forbidden minors for this class of graphs. Open remained how to find a linkless embedding in polynomial time. In the talk we start with discussing an algorithm to find a linkless embedding.Instead of embedding the graph in 3-space, we could also consider mapping properties of certain superstructures of the graph in 3-space, and, indeed, if this superstructure has not the right mapping properties in 3-space, see whether it has the right one in 4-space, etc. Recently, we introduced for a graph G a new graph parameter \sigma(G), which is defined as the smallest d such that superstructures of G have a zero intersection mapping in d-space. The nicest property of this graph parameter is its independence of the superstructure and thus depends on the graph only. For d=2 and d=3, \sigma(G) \leq d if and only if G is outerplanar and planar, respectively. The graphs G with \sigma(G)\leq 4 are exactly those that have a linkless embedding. In the second part of the talk we will discuss this new graph parameter. (This part is joint work with R. Pendavingh.)

We will consider the problem of counting the number T(n,g) of cubic graphs with n edges on a surface of of genus g, and review was is known in the combinatorial community in the past 30 years, what was conjectured in physics 20 years ago, and what was proven last month in joint work with Thang Le and Marcos Marino, using the Riemann-Hilbert analysis of the Painleve equation. No knowledge of physics or analysis is required.

The aim of this talk is to introduce techniques from knot theory into the study of graphs embedded in 3-space. The main characters are hyperbolic geometry and the Jones polynomial. Both have proven to be very successful in studying knots and conjecturally they are intimately related. We show how to extend these techniques to graphs and discuss possible applications. No prior knowledge of knot theory or geometry will be assumed.