Details
| Original language | English |
|---|---|
| Pages (from-to) | 1447-1467 |
| Number of pages | 21 |
| Journal | Algebraic Combinatorics |
| Volume | 6 |
| Issue number | 6 |
| Publication status | Published - 2023 |
Abstract
Keywords
- edge-labeling of graph, free arrangement, graphic arrangement, Hyperplane arrangement, ideal subarrangement, MAT-free arrangement, strongly chordal graph
ASJC Scopus subject areas
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Algebraic Combinatorics, Vol. 6, No. 6, 2023, p. 1447-1467.
Research output: Contribution to journal › Article › Research › peer review
}
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 -