MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling
Tan Nhat Tran, Shuhei Tsujie · Algebraic Combinatorics · 2024
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.