The second Betti number of an edge ideal (Q2721261)

If \(G\) is a graph with vertices \(x_1,\ldots,x_n\) and \(K\) a field then the edge ideal \(I(G)\) of \(G\) is the ideal of \(K[x_1,\ldots,x_n]\) generated by the monomials \(x_ix_j\) such that \(\{x_i,x_j\}\) is an edge of \(G\). NEWLINENEWLINENEWLINEThe authors give an explicit formula for the second initial virtual Betti number of \(I(G)\) in terms of graph-theoretic parameters of \(G\). This is put in context of an open problem whether initial virtual Betti numbers of ideals generated by squarefree monomials are, up to a certain specified degree, independent of \(K\).NEWLINENEWLINEFor the entire collection see [Zbl 0948.00027].