Algebraic study on permutation graphs

TL;DR

The paper studies algebraic properties of permutation graphs via their edge ideals and related invariants. It proves that a permutation graph is Cohen-Macaulay exactly when it is unmixed and vertex decomposable, by translating to a poset via cohesive orders and maximal chains. It also classifies Gorenstein and nearly Gorenstein permutation graphs, derives an explicit formula for the $a$-invariant $a(S/I(G))=\mathrm{im}(G)+\tau(G)-n$, and shows Hilbertianity and bi-Cohen-Macaulay criteria, with consequences for the cover ideal $J(G)$ and its Rees/toric algebras. Overall, the work links permutation-graph structure with homological properties, providing practical criteria and structural descriptions for Cohen-Macaulay and Gorenstein behavior.

Abstract

Let $G$ be a permutation graph. We show that $G$ is Cohen-Macaulay if and only if $G$ is unmixed and vertex decomposable. When this is the case, we obtain a combinatorial description for the $a$-invariant of $G$. Moreover, we characterize the Gorenstein permutation graphs.

Theorems & Definitions (14)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Corollary 2.5
• Corollary 2.6
• proof
• Theorem 2.7
• proof