Mon, August 17, 3pm ET (8pm BST, 7am Tue NZST)
Sophie Spirkl, University of Waterloo
A graph-based introduction to the chromatic symmetric function
Youtube
Abstract:
The chromatic symmetric function is a generalization of the chromatic polynomial, and has been studied extensively, mostly in algebraic combinatorics. I’ll give an introduction to the chromatic symmetric function from a graph theory point of view, including some results and open questions.
Joint work with Logan Crew.
Mon, August 10, 3pm ET (8pm BST, 7am Tue NZST)
Yelena Yuditsky, Ben-Gurion University
Typical structure of hereditary graph families
Youtube
Abstract:
A family of graphs $\cal F$ is hereditary if it is closed under isomorphism and taking induced subgraphs. For example, for a given graph $H$, a hereditary family is the family of all $H$-free graphs, that is graphs without an induced copy of $H$.
Alon, Balogh, Bollobás and Morris showed that for every hereditary family $\cal F$ there exist $\epsilon >0$ and $l\in \mathbb{N}$ such that the number of graphs in $\cal F$ on $n$ vertices is $2^{(1-1/l)n^2/2+o(n^{2-\epsilon})}$. They showed this bound by deriving various structural properties of almost all graphs in $\cal F$. We study and obtain additional structural properties of almost all graphs in some restricted hereditary families. As an application of our results, we prove the existence of an infinite family of counterexamples for the recent Reed-Scott conjecture about the structure of almost all $H$-free graphs.
This is a joint work with Segey Norin.
Mon, August 3, 3pm ET (8pm BST, 7am Tue NZST)
Jim Geelen, University of Waterloo
Towards the excluded-minors for GF(5)-representability
Youtube
Abstract:
I will discuss a strategy towards finding the excluded-minors for GF(5)-representability of matroids. This includes some joint work with Rutger Campbell and Geoff Whittle.
Mon, July 27, 3pm ET (8pm BST, 7am Tue NZST)
Zach Walsh, University of Waterloo
Quadratically Dense Matroids
Youtube
Abstract:
The extremal function of a class of matroids is the function whose value at an integer $n$ is the maximum number of elements of a simple matroid in the class of rank at most $n$. We present a result concerning the role of group-labeled graphs in minor-closed classes of matroids, and then use it to determine the extremal function, for all but finitely many $n$, for the class of complex-representable matroids which exclude a given rank-2 uniform matroid as a minor. This talk will focus on our original motivation, and on the connection between group-labeled graphs and representable matroids.