Algebraic study on permutation graphs

TL;DR

This paper explores algebraic properties of permutation graphs, establishing conditions for Cohen-Macaulay and Gorenstein properties, and providing combinatorial descriptions for algebraic invariants.

Contribution

It characterizes Cohen-Macaulay and Gorenstein permutation graphs and links algebraic properties to combinatorial structures.

Findings

Permutation graphs are Cohen-Macaulay iff they are unmixed and vertex decomposable.

Provides a combinatorial description of the $a$-invariant for such graphs.

Characterizes Gorenstein permutation graphs.

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.