Monday, January 18, 3pm ET (8pm GMT, 9am Tue NZDT)
Kazuhiro Nomoto, University of Waterloo
$I_t$-free, triangle-free binary matroids

Abstract:

For any integer $t \geq 1$, we say that a simple binary matroid is $I_t$-free if no rank-$t$ flats are independent. It is triangle-free if it has no circuit of size $3$. In this talk, we discuss a few problems regarding simple $I_t$-free, triangle-free binary matroids, with some partial results.

Joint work with Peter Nelson.

Monday, January 11, 3pm ET (8pm GMT, 9am Tue NZDT)
Stephan Kreutzer, TU Berlin
Directed tangles

Abstract:

A tangle in a graph $G$ is a consistent orientation of all low-order separations of $G$.
In this way, tangles define the “highly connected” pieces of the graph.

Robertson and Seymour introduced the concept of tangles as part of their graph minor series and proved that the maximal tangles in a graph form a tree structure. More formally, every graph has a tree-decomposition whose pieces correspond to its maximal tangles. This tangle decomposition theorem is one of the essential steps in their proof of the graph minor theorem and has also found many other applications.

Recently, we defined “directed tangles”, the analogue of tangles for digraphs, and showed that again every digraph can be decomposed into a tree of its maximal tangles.

In this talk I will introduce directed tangles and present a proof of the main directed tangle decomposition theorem. I will also focus on the differences between the directed and undirected case and sketch potential applications of the directed tangle decomposition theorem.

The talk is based on joint with with Archontia Giannopoulou, Ken-ichi Kawarabayashi, and O-Joung Kwon.

The open problems session is next week! There are two sessions: Session 1 loosely focused on matroids and Session 2 loosely focused on graphs. Anyone interested in the topics is welcome to join; no registration is required. The sessions will not be recorded, but we will update this page linking to the slide(s) of each presenter.

Monday, December 7, 3pm ET (8pm GMT, 9am Tue NZDT)
Sergey Norin, McGill University
Densities of minor-closed graph classes

Youtube

Abstract:

The density of a sparse graph class is the supremum of the ratio $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ in the class. The densities of graph classes determined by forbidding a single minor have been extensively studied, in part due to connections to the Hadwiger’s conjecture.

We will survey recent results on densities of minor-closed graph classes, including a proof of the conjecture of Eppstein that the densities of minor-closed graph classes are rational, joint with Rohan Kapadia, and joint work with Kevin Hendrey, Bruce Reed, Andrew Thomason and David Wood on densities of classes which forbid a single sparse minor.