  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.
