Georgia Institute of Technology College of Sciences

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^-$. Recently, we proved this entire Kelmans-Seymour conjecture. In this talk, I will give a sketch of our proof, and discuss related problems. This is joint work with Dawei He and Xingxing Yu.

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,\Sigma), where each element x \in X lies in t randomly selected sets of \Sigma, where t \le |X| is an integer parameter. We provide new discrepancy bounds in this case. Specifically, we show that when |\Sigma| \ge |X| the hereditary discrepancy of (X,\Sigma) is with high probability O(\sqrt{t \log t}), matching the Beck-Fiala conjecture upto a \sqrt{\log{t}} factor. Our analysis combines the Lov{\'a}sz Local Lemma with a new argument based on partial matchings.

Flajolet and Odlyzko (1990) derived asymptotic formulae for the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. In this talk, we discuss these multivariate analytic techniques and use them to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities. We will also look at how to apply such formulae to practical problems.

In the Rendezvous problem on the complete graph, two parties are trying to meet at some vertex at the same time, despite starting out with independent random labelings of the vertices. It is well known that the optimal strategy is for one player to wait at any vertex, while the other visits all n vertices in consecutive steps, which guarantees a rendezvous within n steps and takes (n + 1)/2 steps on average. This strategy is very far from being symmetric, however. E. Anderson and R. Weber presented a symmetric algorithm that achieves an expected meeting time of 0.829n, which has been conjectured to be optimal in the symmetric setting. We change perspective slightly: instead of trying to minimize the expected meeting time, we try to maximize the probability of successfully meeting within a specified number of timesteps. In this setting, for all time horizons that are linear in n, the Anderson-Weber strategy has a constant probability of failure. Surprisingly, we show that this is not optimal: there exists a different symmetric strategy that almost surely guarantees meeting within 4n timesteps. This bound is tight, in that the factor 4 cannot be replaced by any smaller constant. Our strategy depends on the construction of a new kind of combinatorial object that we dub”rendezvous code.”On the positive side, for T < n, we show that the probability of meeting within T steps is indeed (approximately) maximized by the Anderson-Weber strategy. Our results imply new lower bounds on the expected meeting time for any symmetric strategy, which establishes an asymptotic difference between the best symmetric and asymmetric strategies. Finally, we examine the symmetric rendezvous problem on other vertex-transitive graphs.