Proof of a conjecture of Bergeron , Ceballos and Labbé

The reduced expressions for a given element w of a Coxeter group (W, S) can be regarded as the vertices of a directed graphR (w); its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression −→a to a reduced expression −→ b when −→ b is obtained from −→a by replacing a contiguous subword of the form stst · · · (for some distinct s, t ∈ S) by tsts · · · (where both subwords have length ms,t, the order of st ∈ W). We prove a strong bipartiteness-type result for this graph R (w): Not only does every cycle of R (w) have even length; actually, the arcs of R (w) can be colored (with colors corresponding to the type of braid moves used), and to every color c corresponds an “opposite” color cop (corresponding to the reverses of the braid moves with color c), and for any color c, the number of arcs in any given cycle of R (w) having color in {c, cop} is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé.