# Milo Orlich: Asymptotic results on the regularity of edge ideals of graphs via critical graphs

**Time: **
Mon 2022-12-12 15.00 - 16.00

**Location: **
Zoom

**Video link: **
Meeting ID: 617 3654 7173

**Participating: **
Milo Orlich, Aalto Universit

### Abstract

To any graph G one can associate its edge ideal. One of the most famous results in combinatorial commutative algebra, Hochster's formula, describes the Betti numbers of the edge ideal in terms of combinatorial information on the graph G. More explicitly, each specific Betti number is given in terms of the occurrence of certain induced subgraphs in G. The machinery of critical graphs, relatively recently introduced by Balogh and Butterfield, deals with characterizing asymptotically the structure of graphs based on their induced subgraphs. In a joint work with Alexander Engström, "The regularity of almost all edge ideals", we apply these techniques to Betti numbers and regularity of edge ideals. We introduce parabolic Betti numbers, which constitute a non-trivial portion of the Betti table. One of our main results is that, fixed a parabolic Betti number on row r of the Betti table, for almost all graphs with that Betti number equal to zero the regularity of the edge ideal is r-1, and the graph can be covered with r-2 cliques and one independent set.