YouTube recording: https://www.youtube.com/watch?v=veVUV1Ti0Wc

Time: Thursday, Mar 2, 3pm EST (8pm GMT, 9am Fri NZDT)
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Raphael Steiner, ETH Zürich
Title: Coloring hypergraphs with excluded minors

Abstract: Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly (t−1)-colorable. The purpose of this work is to demonstrate that a natural extension of Hadwiger’s problem to hypergraph coloring exists, and to derive some first partial results and applications. Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_1$ is a minor of a hypergraph $H_2$, if a hypergraph isomorphic to $H_1$ can be obtained from $H_2$ via a finite sequence of vertex- and hyperedge-deletions, and hyperedge contractions. We first show that a weak extension of Hadwiger’s conjecture to hypergraphs holds true: For every positive integer t, there exists a finite (smallest) integer h(t) such that every hypergraph with no $K_t$-minor is h(t)-colorable, and we prove $$\lceil 3/2(t-1) \rceil \le h(t) \le 2g(t)$$ where g(t) denotes the maximum chromatic number of graphs with no $K_t$-minor. Using the recent result by Delcourt and Postle that $g(t) = O(t \log\log t)$, this yields $h(t) = O(t\log\log t)$. We further conjecture that $h(t) = \lceil 3/2(t-1) \rceil$, i.e., that every hypergraph with no $K_t$-minor is $\lceil 3/2(t-1) \rceil$-colorable for all t, and prove this conjecture for all hypergraphs with independence number at most 2. By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as:
-graphs of chromatic number $Ckt\log\log t$ contain $K_t$-minors with k-edge-connected branch-sets,
-graphs of chromatic number $Cqt\log\log t$ contain $K_t$-minors with modulo-q-connected branch sets,
-by considering cycle hypergraphs of digraphs we recover known results on strong minors in digraphs of large dichromatic number as special cases.

YouTube recording: https://www.youtube.com/watch?v=6hmhSYado9Q

Time: Thursday, Feb 16, 3pm EST (8pm GMT, 9am Fri NZDT)
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Relinde Jurrius, Netherlands Defense Academy
Title: The combinatorial derived matroid
Abstract: The question of “dependencies between dependencies” in matroids has been raised in the 1960’s by Rota and Crapo, in various formulations (here is a nice overview elsewhere on this blog: matroidunion.org/?p=2628). For a matroid $M$, the derived matroid is a matroid that has as ground set the set of (co)circuits of $M$. Over time, various definitions of a derived matroid have been proposed. However, none of them completely solved the question of Rota and Crapo: these definitions depend on a representation, for example, or are not unique, or don’t always exist.
In a recent paper with Olga Kuznetsova and Ragnar Freij-Hollanti, we proposed a definition of a derived matroid that is purely combinatorial. The combinatorial derived matroid is defined via the rank function of $M$, and via an operation that resembles a closure operation on the collection of dependent sets. It is therefore uniquely defined for any matroid. In this talk I will motivate this definition and discuss some examples and desirable properties of this definition. Time permitting we will discuss open questions and links to previous definitions.