Tuesday, Feb 1, 3pm ET (8pm GMT, 9am Wed NZDT)
Matthew Coulson, University of Waterloo
Strong components of the directed configuration model

Abstract:

We study the behaviour of the largest components of the directed configuration model in the barely subcritical regime. We show that with high probability all strongly connected components in this regime are either cycles or isolated vertices and give an asymptotic distribution of the size of the kth largest cycle. This gives a configuration model analogue of a result of Luczak and Seierstad for the binomial random digraph.

Tuesday, Jan 25, 3pm ET (8pm GMT, 9am Wed NZDT)
Archontia Giannopoulou, University of Athens
A Matching Theoretic Flat Wall Theorem

Abstract:

One of the key theorems in Graph Minors is the Flat Wall Theorem which asserts the existence of a large wall under certain conditions. In this talk, we discuss about graphs with perfect matchings and their relationship with digraphs. Our main focus is on a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. The tight relationship between structural digraph theory and matching theory that allows us to obtain the aforementioned version of Flat Wall Theorem further allow us to deduce a Flat Wall Theorem for digraphs which substantially differs from a previously established directed variant of this theorem.

Joint work with Sebastian Wiederrecht.

Tuesday, Jan 18, 3pm ET (8pm GMT, 9am Wed NZDT)
Mehtaab Sawhney, MIT
Enumerating Matroids and Linear Spaces

Abstract:

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: there are exact formulas for enumeration of rank-1 and rank-2 matroids, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area.