Cohen--Macaulayness of $\mathrm{Inc}(\mathbb{N})$-Invariant Chains of Edge Ideals

Abstract: Let $(G_n)<em>{n\ge n_0}$ be a family of graphs on $[n]$ whose edge ideals $I_n\subseteq R_n=k[x_1,\dots,x_n]$ form an $\mathrm{Inc}(\mathbb{N})$-invariant chain, $$I</em>{n+r}=\mathrm{Inc}(\mathbb{N})<em>{n,n+r}(I_n)$$ for $n\ge n_0,\ r\ge0$. We determine which pairs $(n,r)$ make $R</em>{n+r}/I_{n+r}$ Cohen--Macaulay, for five classical families: line graphs $L_n$, complements of line graphs $L_n<sup>c$, complete graphs $K_n$, cyclic graphs $C_n$, and complements of cyclic graphs $C_n<sup>c$. For line graphs we give the generators of $I_{n+r}$, the height and Krull dimension of $R_{n+r}/I_{n+r}$, and a complete classification: for $n\ge5$, $R_{n+r}/I_{n+r}$ is Cohen--Macaulay if and only if $r=n-4$. For complements of line graphs, complete graphs, and complements of cyclic graphs, we show the chain is Cohen--Macaulay unconditionally, for every $r\ge0$; the first and third arise from the same underlying phenomenon, in which the $\mathrm{Inc}(\mathbb{N})$-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove $$\mathrm{Inc}(\mathbb{N})<em>{n,n+r}(I(C_n))=I(K</em>{n+r})$$ once $r\ge n-3$, giving Cohen--Macaulayness in this range, and conjecture -- with supporting computational and structural evidence -- a complete classification: Cohen--Macaulayness holds if and only if $r\ge\lfloor(n-4)/2\rfloor$ and $r\ne n-4$.