Title. Universal Graph Product Structures
Speaker. David R. Wood

Abstract. This talk will survey recent results that describe graphs as subgraphs of products of simpler graphs. The starting point is the following theorem: every planar graph is a subgraph of the strong product of some treewidth 7 graph and a path. This result has been the key to solving several open problems, for example, regarding the queue-number and nonrepetitive chromatic number of planar graphs. The result generalises to provide a universal graph for planar graphs. In particular, if $T^7$ is the universal treewidth 7 graph (which is explicitly defined), then every countable planar graph is a subgraph of the strong product of $T^7$ and the infinite 1-way path. This result generalises in various ways for many sparse graph classes: graphs embeddable on a fixed surface, graphs excluding a fixed minor, map graphs, etc. For example, we prove that for every fixed graph $X$, there is an explicitly defined countable graph $G$ that contains every countable $X$-minor-free graph as a subgraph, and $G$ has several desirable properties such as every $n$-vertex subgraph of $G$ has a $O(\sqrt{n})$-separator. On the other hand, as a lower bound we strengthen a theorem of Pach (1981) by showing that if a countable graph $G$ contains every countable planar graph, then $G$ must contain an infinite complete graph as a minor.