The regularity of almost all edge ideals

TL;DR

This paper explores the structure of edge ideals of graphs, revealing that almost all such graphs with certain Betti number properties can be partitioned into specific cliques and independent sets, with implications for their regularity.

Contribution

It transfers the graph-theoretic paradigm of common structure in almost all graphs to the algebraic setting of edge ideals, identifying a new class of Betti numbers called parabolic Betti numbers.

Findings

Almost all graphs with certain Betti number conditions can be partitioned into r-2 cliques and one independent set.

For these graphs, the regularity of the edge ideal is r-1.

The study introduces the concept of parabolic Betti numbers within the Betti table.

Abstract

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield. We consider edge ideals $I_G$ of graphs and their Betti numbers. The numbers of the form $\beta_{i,2i+2}$ constitute the "main diagonal" of the Betti table. It is well known that any Betti number $\beta_{i,j}(I_G)$ below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let $\beta_{i,j}$ be a parabolic Betti number on the $r$-th row of the Betti table, for $r\ge3$. Our main results state that almost all graphs $G$ with $\beta_{i,j}(I_G)=0$ can be partitioned into $r-2$ cliques and one independent set, and in particular for almost all graphs $G$ with $\beta_{i,j}(I_G)=0$ the regularity of $I_G$ is $r-1$.