Signed-eliminable graphs and free multiplicities on the braid arrangement

TL;DR

This paper introduces bicolor-eliminable graphs to classify free multiplicities on the braid arrangement, extending Stanley's theory, and proves part of Athanasiadis's conjecture relating freeness to directed graphs.

Contribution

It generalizes Stanley's classification of free graphic arrangements to bicolor graphs and provides a complete classification of certain free multiplicities on the braid arrangement.

Findings

Complete classification of free multiplicities via bicolor-eliminable graphs

Extension of Stanley's chordal graph theory to bicolor graphs

Partial proof of Athanasiadis's conjecture on deformation freeness

Abstract

We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.