Proof of a conjecture of Bergeron, Ceballos and Labb\'e

The reduced expressions for a given element $w$ of a Coxeter group $(W, S)$ can be regarded as the vertices of a directed graph $\mathcal{R}(w)$; 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 $s, t$ in $S$) by $tsts...$ (where both subwords have length $m_{s, t}$, the order of $st$ in $W$). We prove a strong bipartiteness-type result for this graph $\mathcal{R}(w)$: Not only does every cycle of $\mathcal{R}(w)$ have even length; actually, the arcs of $\mathcal{R}(w)$ can be colored (with colors corresponding to the type of braid moves used), and to every color $c$ corresponds an "opposite" color $c^{\operatorname{op}}$ (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 $\mathcal{R}(w)$ having color in $\left\{c, c^{\operatorname{op}}\right\}$ is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labb\'e. We state further conjectural extensions.