Georgia Institute of Technology College of Sciences

We use K_4^- to denote the graph obtained from K_4 by removing an edge,and use TK_5 to denote a subdivision of K_5. Let G be a 5-connected nonplanar graph and {x_1, x_2, y_1, y_2} \subseteq V (G) such that G[{x_1,x_2, y_1, y_2}] = K_4^- with y_1y_2 \in E(G). Let w_1,w_2,w_3 \in N(y_2)- {x_1,x_2} be distinct. We show that G contains a TK_5 in which y_2 is not a branch vertex, or G - y_2 contains K_4^-, or G has a special 5-separation, or G' := G - {y_2v : v \in {w_1,w_2,w_3, x_1, x_2}} contains TK_5.In this talk, we will obtain a substructure in G' and several additional paths in G', and then use this substructure to find the desired TK_5.

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^-. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. We will talk about special 5-separations and 6-separations whose cut contains a triangle. Results will be used in subsequently to prove the Kelmans-Seymour conjecture.

The goal of this talk is to show recent advances regarding two important mathematical problems. The first one can be straightforwardly formulated in a graph theory language, but can be possibly applied in other fields. The second one was motivated by machine learning applications, but leads to graph theory techniques. The celebrated open conjecture of Erdos and Hajnal from 1989 states that families of graphs not having some given graph H as an induced subgraph contain polynomial-size cliques/stable sets (in the undirected setting) or transitive subsets (in the directed setting). Recent techniques developed over last few years provided the proof of the conjecture for new infinite classes of graphs (in particular the first infinite class of prime graphs). Furthermore, they gave tight asymptotics for the Erdos-Hajnal coefficients for many classes of prime tournaments as well as the proof of the conjecture for all but one tournament on at most six vertices and the proof of the weaker version of the conjecture for trees on at most six vertices. In this part of the talk I will summarize these recent achievements. Structured non-linear graph-based hashing is motivated by applications in neural networks, where matrices of linear projections are constrained to have a specific structured form. This drastically reduces the size of the model and speeds up computations. I will show how the properties of the underlying graph encoding correlations between entries of these matrices (such as its chromatic number) imply the quality of the entire non-linear hashing mechanism. Furthermore, I will explain how general structured matrices that very recently attracted researchers’ attention naturally lead to the underlying graph theory description.

A set F of graphs has the Erdos-Posa property if there exists a function f such that every graph either contains k disjoint subgraphs each isomorphic to a member in F or contains at most f(k) vertices intersecting all such subgraphs. In this talk I will address the Erdos-Posa property with respect to three closely related graph containment relations: minor, topological minor, and immersion. We denote the set of graphs containing H as a minor, topological minor and immersion by M(H),T(H) and I(H), respectively. Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa property if and only if H is planar. And they left the question for characterizing H in which T(H) has the Erdos-Posa property in the same paper. This characterization is expected to be complicated as T(H) has no Erdos-Posa property even for some tree H. In this talk, I will present joint work with Postle and Wollan for providing such a characterization. For immersions, it is more reasonable to consider an edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs and covering them by edges. I(H) has no this edge-variant of the Erdos-Posa property even for some tree H. However, I will prove that I(H) has the edge-variant of the Erdos-Posa property for every graph H if the host graphs are restricted to be 4-edge-connected. The 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.