Generic Subideals of Graph Ideals and Free Resolutions

For a graph of an n-cycle ∆ with Alexander dual ∆, we study the free resolution of a subideal G(n) of the Stanley-Reisner ideal I∆∗ . We prove that if G(n) is generated by 3 generic elements of I∆∗ , then the second syzygy module of G(n) is isomorphic to the second syzygy module of (x1, x2, . . . , xn). A result of Bruns shows that there is always a 3-generated ideal with this property. We show that it can be chosen to have a particularly nice form.