Leírás
This talk will be divided into two independent parts.
The first part will be about the dimer model which is a statistical mechanics model that studies random perfect matchings on graphs. It was introduced in the 1960s, when the partition functions and correlations were exactly computed by Kasteleyn, Temperley, and Fisher. Over the past few decades, significant progress has been made in the bipartite case, particularly concerning spectral curves, integrable systems, and limit shapes. I will present recent results on the Newton polygons associated with the characteristic polynomials arising from the dimer model of certain families of non-bipartite graphs.
In the second part, I will discuss a joint work with Balázs Keszegh. The saturation problem asks for the minimum number of edges in a graph H on n vertices, which does not contain G as a subgraph but adding any
missing edge to H creates a copy of G. This problem can be studied for various combinatorial objects. For graphs, 0-1 and vertex-ordered graphs it was shown that their saturation functions are either bounded or
linear. We study the saturation functions of edge-ordered graphs and we show that there is no such dichotomy by providing examples of edge-ordered graphs with superlinear saturation functions.