Georgia Institute of Technology College of Sciences

Consider a graph G and a specified subset A of vertices. An A-path is a path with both ends in A and no internal vertex in A. Gallai showed that there exists a min-max formula for the maximum number of pairwise disjoint A-paths. More recent work has extended this result, considering disjoint A-paths which satisfy various additional properties. We consider the following model. We are given a list of {(s_i, t_i): 0< i < k} of pairs of vertices in A, consider the question of whether there exist many pairwise disjoint A-paths P_1,..., P_t such that for all j, the ends of P_j are equal to s_i and t_i for some value i. This generalizes the disjoint paths problem and is NP-hard if k is not fixed. Thus, we cannot hope for an exact min-max theorem. We further restrict the question, and ask if there either exist t pairwise disjoint such A-paths or alternatively, a bounded set of f(t) vertices intersecting all such paths. In general, there exist examples where no such function f(t) exists; we present an exact characterization of when such a function exists. This is joint work with Daniel Marx.

By the 4-color theorem, every planar graph on n vertices has an independent set of size at least n/4. Finding a simple proof of this fact is a long-standing open problem. Furthermore, no polynomial-time algorithm to decide whether a planar graph has an independent set of size at least (n+1)/4 is known. We study the analogous problem for triangle-free planar graphs. By Grotzsch' theorem, each such graph on n vertices has an independent set of size at least n/3, and this can be easily improved to a tight bound of (n+1)/3. We show that for every k, a triangle-free planar graph of sufficiently large tree-width has an independent set of size at least (n+k)/3, thus giving a polynomial-time algorithm to decide the existence of such a set. Furthermore, we show that there exists a constant c < 3 such that every planar graph of girth at least five has an independent set of size at least n/c.Joint work with Matthias Mnich.

We prove the dcdc conjecture in a class of lean fork graphs, argue that this class is substantial and show a path towards the complete solution. Joint work with Andrea Jimenez.

Since the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs over 20 years ago, there has been a lot of effort by many researchers to extend the theory to hypergraphs. I will present some of this history, and then describe our recent results that provide such a generalization and unify much of the previous work. One key new aspect in the theory is a systematic study of hypergraph eigenvalues first introduced by Friedman and Wigderson. This is joint work with John Lenz.