TY - JOUR

T1 - MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

AU - Tran, Tan Nhat

AU - Tsujie, Shuhei

N1 - Funding Information: The first author was supported by JSPS Research Fellowship for Young Scien- tists Grant Number 19J12024 at Hokkaido University and a postdoctoral fellowship of the Alexander von Humboldt Foundation at Ruhr-Universität Bochum.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - edge-labeling of graph

KW - free arrangement

KW - graphic arrangement

KW - Hyperplane arrangement

KW - ideal subarrangement

KW - MAT-free arrangement

KW - strongly chordal graph

UR - http://www.scopus.com/inward/record.url?scp=85165721441&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2204.08878

DO - 10.48550/arXiv.2204.08878

M3 - Article

VL - 6

SP - 1447

EP - 1467

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

IS - 6

ER -