Georgia Institute of Technology College of Sciences

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

Please note the unusual time/day.

An n-lift of a graph G is a graph from which there is an n-to-1 covering map onto G. Amit, Linial, and Matousek (2002) raised the question of whether the chromatic number of a random n-lift of K_5 is concentrated on a single value. We consider a more general problem, and show that for fixed d ≥ 3 the chromatic number of a random lift of K_d is (asymptotically almost surely) either k or k+1, where k is the smallest integer satisfying d < 2k log k. Moreover, we show that, for roughly half of the values of d, the chromatic number is concentrated on k. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random n-lifts of G, for any fixed d-regular graph G. (This is joint work with JD Nir.)

We shall discuss a recent paper of Wanless and Wood (arXiv:2008.00775), which proves a Lovász Local Lemma type result using inductive counting arguments.
For example, in the context of hypergraph colorings, under LLL-type assumptions their result typically gives exponentially many colorings (usually more than the textbook proof of LLL would give).
We will present a probabilistic proof of the Wanless-Wood result, and discuss some applications to k-SAT, Ramsey number lower bounds, and traversals, as time permits.

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. 
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.  
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. 
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.