Projective dimension of weakly chordal graphic arrangements

TL;DR

This paper extends classical results on graphic arrangements by characterizing when their module of logarithmic derivations has projective dimension at most one, linking it to the graph being weakly chordal.

Contribution

It proves that the projective dimension of the module of logarithmic derivations is at most one if and only if the graph and its complement are weakly chordal.

Findings

Projective dimension at most one for logarithmic derivations corresponds to weakly chordal graphs.

Extends classical chordal graph results to weakly chordal graphs.

Provides a new algebraic characterization of weakly chordal graphs.

Abstract

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.