Edge ideals with almost maximal finite index and their powers

Abstract

A graded ideal in K[x1, . . . , xn], K a field, is said to have almost maximal finite index if all steps of its minimal free resolution are linear except for the last two steps. In this paper we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals, unlike the general case, the property of having almost maximal finite index does not depend on the characteristic of K. We also compute the non-linear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal I s has a linear resolution for s ≥ 2 if and only if the complementary graph Ḡ does not contain induced cycles of length 4.