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.
Splittable ideals and the resolutions of monomial ideals.Journal of Algebra, 309(1):405–425
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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.