Cohen–Macaulayness classification for cyclic-graph invariant chains

Determine whether, for every integer n≥6 and r≥0, the ideal I_{n+r}:=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I(C_n)) is Cohen–Macaulay if and only if r≥\lfloor(n-4)/2\rfloor and r\ne n-4.

Background

The paper studies chains of edge ideals generated from a graph on [n] by applying all strictly increasing injections into [n+r]. For the cyclic graph C_n, the authors prove that the resulting ideal becomes the edge ideal of the complete graph K_{n+r} whenever r≥n-3, which establishes Cohen–Macaulayness in that range.

Computational and structural evidence suggests that the complete-graph threshold is not optimal. The proposed classification asserts Cohen–Macaulayness beginning already at r=\lfloor(n-4)/2\rfloor, except at the isolated value r=n-4. At that exceptional value, the independent pairs form a disjoint union of antipodal edges, making the independence complex disconnected and therefore preventing Cohen–Macaulayness.