A new upper bound for the Asymptotic Dimension of RACGs

Let $W_\Gamma$ be the Right-Angled Coxeter group with defining graph $\Gamma$. We show that the asymptotic dimension of $W_\Gamma$ is smaller than or equal to $dim_{CC}(\Gamma)$, the clique-connected dimension of the graph. As a corollary we show that $W_\Gamma$ is virtually free if and only if $dim_{CC}(\Gamma)=1$.