MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

TL;DR

This paper characterizes MAT-free graphic arrangements, showing they correspond exactly to strongly chordal graphs, thus linking algebraic freeness properties with specific graph-theoretic structures.

Contribution

It provides a precise characterization of MAT-freeness for type A Weyl subarrangements, establishing a direct correspondence with strongly chordal graphs.

Findings

MAT-free graphic arrangements correspond to strongly chordal graphs

MAT-freeness is closed under localization for graphic arrangements

The characterization answers an open question by Cuntz-Mücksch

Abstract

Ideal subarrangements of a Weyl arrangement are proved to be free by the multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao (2016). They form a significant class among Weyl subarrangements that are known to be free so far. The concept of MAT-free arrangements was introduced recently by Cuntz-M{\"u}cksch (2020) to capture a core of the MAT, which enlarges the ideal subarrangements from the perspective of freeness. The aim of this paper is to give a precise characterization of the MAT-freeness in the case of type $A$ Weyl subarrangements (or graphic arrangements). It is known that the ideal and free graphic arrangements correspond to the unit interval and chordal graphs respectively. We prove that a graphic arrangement is MAT-free if and only if the underlying graph is strongly chordal. In particular, it affirmatively answers a question of Cuntz-M{\"u}cksch that MAT-freeness is closed under taking localization in the case of graphic arrangements.