ALGEBRAIC COMBINATORICS

no. 6
MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling
Tran, Tan N.1; Tsujie, Shuhei2
1 Leibniz Universität Hannover Fakultät für Mathematik und Physik, Institut für Algebra, Zahlentheorie und Diskrete Mathematik Welfengarten 1, D-30167 Hannover Germany
2 Hokkaido University of Education Department of Mathematics Asahikawa, Hokkaido 070-8621 Japan
Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1447-1467.

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.319
Classification: 52C35, 13N15, 05C78
Keywords: Hyperplane arrangement, free arrangement, MAT-free arrangement, ideal subarrangement, graphic arrangement, strongly chordal graph, edge-labeling of graph
Tran, Tan N. 1; Tsujie, Shuhei 2

1 Leibniz Universität Hannover Fakultät für Mathematik und Physik, Institut für Algebra, Zahlentheorie und Diskrete Mathematik Welfengarten 1, D-30167 Hannover Germany
2 Hokkaido University of Education Department of Mathematics Asahikawa, Hokkaido 070-8621 Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Tran, Tan N.; Tsujie, Shuhei. MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1447-1467. doi : 10.5802/alco.319. https://alco.centre-mersenne.org/articles/10.5802/alco.319/

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