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. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.For a graph G, let IG denote its edge ideal, the monomial ideal generated by xixj for every edge ij of G. In this paper we study the graded Betti numbers of IG, which are combinatorial invariants that measure the complexity of a minimal free resolution of IG. The Betti numbers of the form βi,2i+2 constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number 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 βi,j be a parabolic Betti number on the r-th row of the Betti table, for r≥3. We prove that almost all graphs G with βi,j(IG)=0 can be partitioned into r−2 cliques and one independent set. In particular, for almost all graphs G with βi,j(IG)=0, the regularity of IG is r−1.