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

authored by

Tan Nhat Tran, Shuhei Tsujie

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–Mücksch (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 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–Mücksch that MAT-freeness is closed under taking localization in the case of graphic arrangements.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Hokkaido University of Education

Type

Article

Journal

Algebraic Combinatorics

Volume

6

Pages

1447-1467

No. of pages

21

Publication date

2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Electronic version(s)

https://doi.org/10.48550/arXiv.2204.08878 (Access: Open)
https://doi.org/https://alco.centre-mersenne.org/articles/10.5802/alco.319/ (Access: Open)