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A fruitful contemporary paradigm in graph theory is that almost all graphsthat do not contain a certain subgraph have common structural characteristics.The "almost" is crucial, without it there is no structure. In this paper wetransfer this paradigm to commutative algebra and make use of deep graphtheoretic results. A key tool are the critical graphs introduced by Balogh andButterfield.  We consider edge ideals $I_G$ of graphs and their Betti numbers. The numbersof 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 (orequivalently, to the left of) this diagonal is always zero. We identify acertain "parabola" inside the Betti table and call parabolic Betti numbers theentries of the Betti table bounded on the left by the main diagonal and on theright 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 almostall graphs $G$ with $\beta_{i,j}(I_G)=0$ can be partitioned into $r-2$ cliquesand one independent set, and in particular for almost all graphs $G$ with$\beta_{i,j}(I_G)=0$ the regularity of $I_G$ is $r-1$.

The regularity of almost all edge ideals