Removing a suitable Hamiltonian cycle from generalized Andrásfai graphs GA(t,k) yields edge ideals with regularity t+2, projective dimension t(k-2), and a generalized special Betti diagram.
First nonlinear syzygies of ideals associated to graphs.Communica- tions in Algebra, 37(6):1921–1933
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Generalized Andr\'asfai graphs and special Betti diagrams of edge ideals
Removing a suitable Hamiltonian cycle from generalized Andrásfai graphs GA(t,k) yields edge ideals with regularity t+2, projective dimension t(k-2), and a generalized special Betti diagram.