Signed-eliminable graphs and free multiplicities on the braid arrangement

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.

19 pages. In version 3, the errors in the statement and the proof of Theorem 0.3 are corrected in the Appendix