发表机构
Lahore University of Management Sciences（拉合尔管理科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对五类经典图族，研究Inc(ℕ)-不变边理想链对应的商环的Cohen--Macaulay性，给出线图的完整分类，证明三类图的链无条件Cohen--Macaulay，确定循环图的Cohen--Macaulay范围并提出完整分类猜想。
AI 中文摘要
设$(G_n)_{n\ge n_0}$是$[n]$上的图族，其边理想$I_n\subseteq R_n=k[x_1,\dots,x_n]$构成一个$\mathrm{Inc}(\mathbb{N})$-不变链，满足对$n\ge n_0,\\ r\ge0$有$I_{n+r}=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I_n)$。我们针对五类经典图族，确定了哪些$(n,r)$对使得$R_{n+r}/I_{n+r}$是Cohen--Macaulay的，这五类图族分别是线图$L_n$、线图的补图$L_n^c$、完全图$K_n$、循环图$C_n$以及循环图的补图$C_n^c$。对于线图，我们给出了$I_{n+r}$的生成元、$R_{n+r}/I_{n+r}$的高度和Krull维数，并得到完整分类：当$n\ge5$时，$R_{n+r}/I_{n+r}$是Cohen--Macaulay的当且仅当$r=n-4$。对于线图的补图、完全图以及循环图的补图，我们证明该链是无条件Cohen--Macaulay的，即对每个$r\ge0$均成立；其中第一类和第三类图源于相同的 underlying 现象，即$\mathrm{Inc}(\mathbb{N})$-不变链在每一步都重现了原始图本身。对于循环图，我们证明当$r\ge n-3$时，$\mathrm{Inc}(\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$，从而在该范围内得到Cohen--Macaulay性，并结合计算和结构证据提出完整分类猜想：Cohen--Macaulay性成立当且仅当$r\ge\lfloor(n-4)/2\rfloor$且$r\ne n-4$。
英文摘要
Let $(G_n)_{n\ge n_0}$ 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, $I_{n+r}=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I_n)$ for $n\ge n_0,\ r\ge0$. We determine which pairs $(n,r)$ make $R_{n+r}/I_{n+r}$ Cohen--Macaulay, for five classical families: line graphs $L_n$, complements of line graphs $L_n^c$, complete graphs $K_n$, cyclic graphs $C_n$, and complements of cyclic graphs $C_n^c$. 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 $\mathrm{Inc}(\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$ 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 $r\ne n-4$.
Comments18 pages
