Betti numbers of chordal graphs and f-vectors of simplicial complexes

Let G be a chordal graph and I(G) its edge ideal. Let β(I(G))=(β0,β1,…,βp) denote the Betti sequence of I(G), where βi stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex Δ of dimension p whose f-vector f(Δ)=(f0,f1,…,fp) coincides with β(I(G)).