Cohen--Macaulayness of Inc(ℕ)-Invariant Chains of Edge Ideals
Abstract
Let (Gn)n≥ n0 be a family of graphs on [n] whose edge ideals In⊆ Rn=k[x1,…,xn] form an Inc(ℕ)-invariant chain, In+r=Inc(ℕ)n,n+r(In) for n≥ n0, r≥0. We determine which pairs (n,r) make Rn+r/In+r Cohen--Macaulay, for five classical families: line graphs Ln, complements of line graphs Lnc, complete graphs Kn, cyclic graphs Cn, and complements of cyclic graphs Cnc. For line graphs we give the generators of In+r, the height and Krull dimension of Rn+r/In+r, and a complete classification: for n≥5, Rn+r/In+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≥0; the first and third arise from the same underlying phenomenon, in which the Inc(ℕ)-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove Inc(ℕ)n,n+r(I(Cn))=I(Kn+r) once r≥ 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≥⌊(n−4)/2⌋ and r≠ n−4.
Related
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BibTeX
@misc{anwar2026cohenmacaulayness,
title = {{Cohen--Macaulayness of $\mathrmInc(\mathbbN)$-Invariant Chains of Edge Ideals}},
author = {Imran Anwar and Mughees Ghayas and Azhar Javed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03566},
eprint = {2609.03566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03566},
note = {CC0},
}
RIS
TY - UNPB
TI - Cohen--Macaulayness of Inc(ℕ)-Invariant Chains of Edge Ideals
AU - Imran Anwar
AU - Mughees Ghayas
AU - Azhar Javed
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03566
UR - https://arxiv.org/abs/2609.03566
AB - Let (G_n)_n≥ n_0 be a family of graphs on [n] whose edge ideals I_n⊆ R_n=k[x_1,…,x_n] form an Inc(ℕ)-invariant chain, I_n+r=Inc(ℕ)_n,n+r(I_n) for n≥ n_0, r≥0. 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≥5, 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≥0; the first and third arise from the same underlying phenomenon, in which the Inc(ℕ)-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove Inc(ℕ)_n,n+r(I(C_n))=I(K_n+r) once r≥ 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≥⌊(n−4)/2⌋ and r≠ n−4.
ER -