Online talk: Johannes Carmesin

Monday, November 30, 3pm ET (8pm GMT, 9am Tue NZDT)
Johannes Carmesin, University of Birmingham
Matroids and embedding graphs in surfaces
Given a graph, how do we construct a surface so that the graph embeds in that surface in an optimal way? Thomassen showed that for minimum genus as optimality criterion, this problem would be NP-hard. Instead of minimum genus, here we use local planarity — and provide a polynomial algorithm.
Our embedding method is based on Whitney’s trick to use matroids to construct embeddings in the plane. Consequently we obtain a characterisation of the graphs admitting locally planar embeddings in surfaces in terms of a certain matroid being co-graphic.

Online talk: Sarah Allred

Monday, November 23, 3pm ET (8pm GMT, 9am Tue NZDT)
Sarah Allred, LSU
Unavoidable Induced Subgraphs of Large 2-connected Graphs
It is well known that, for every positive integer $r$, every sufficiently large connected graph contains an induced subgraph isomorphic to one of the following: a large complete graph, a large star, and a long path.  We prove an analogous result for 2-connected graphs.  In particular, we show that every sufficiently large 2-connected graph contains an induced subgraph isomorphic to one of the following: a complete graph, a subdivision of $K_{2,r}$ with possibly an edge joining the two vertices of degree $r$, and a graph that has a well-described ladder structure.
This is joint work with Guoli Ding and Bogdan Oporowski. 

Online talk: Jakob Kneip

Monday, November 16, 3pm ET (8pm GMT, 9am Tue NZDT)
Jakob Kneip, Universität Hamburg
Tangles are decided by weighted vertex sets
Slides (view pdf)

Tangles are a way to capture clusters or highly cohesive regions in graphs: a tangle in a graph declares for each low-order separation one of the two sides of that separation as ‘big’, and does so in such a way that no three ‘small’ sides together cover the graph. Most types of clusters in graphs — for instance clique- or grid-minors — give rise to such a tangle by declaring as ‘big’ for each separation the side which contains the majority of the vertices of that cluster.

An open question in the theory of tangles is whether a converse to this holds: given a tangle in a graph, can we find a set of vertices such that the ‘big’ side of each separation is precisely the side which contains the majority of the set’s vertices? A positive answer to this would validate the intuition that the sides declared as ‘big’ by a tangle are in fact big in some sense.

In joint work with Christian Elbracht and Maximilian Teegen we show that a fractional version of this is true: given a tangle we construct a weight function on the vertices such the ‘big’ side of each separation is exactly the side with higher total weight. We also show that our result extends to tangles of hypergraphs as well as to edge tangles, which are based on cuts rather than vertex separations. Curiously though one cannot achieve both extensions simultaneously: there are edge tangles of hypergraphs that are not defined by such a weight function on the vertices.

Online talk: James Davies

Monday, November 9, 3pm ET (8pm GMT, 9am Tue NZDT)
James Davies, University of Waterloo
Geometric intersection graphs with large girth and chromatic number

We prove that both the classes of intersection graphs of axis-aligned boxes in $\mathbb{R}^3$ and intersection graphs of straight line segments in $\mathbb{R}^3$ contain graphs with arbitrarily large girth and chromatic number.