Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

A fast new algorithm for weak graph regularity

We provide a deterministic algorithm that finds, in ϵ-O(1) n 2 time, an ϵ-regular Frieze-Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ϵ-O(1) many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of H in an n-vertex graph G up to an additive error of at most ϵn v(H), in time ϵ-O H(1) n 2National Science Foundation (U.S.). Postdoctoral Fellowship (Award DMS 1705204)National Science Foundation (U.S.) (Award DMS 1362326