The regularity of almost all edge ideals

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 $r-1$.