Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

Abstract

The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $A$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. This can be regarded as a counterpart of the characterization by Stanley and Edelman–Reiner of free and supersolvable graphic arrangements in terms of chordal graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.