Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

If $A_\Gamma$ ($W_\Gamma$) is the Artin (Coxeter) group with defining graph $\Gamma$ we denote by $Sim(\Gamma)$ the number of vertices of the largest clique in $\Gamma$. We show that $asdimA_\Gamma \leq Sim(\Gamma)$, if $Sim(\Gamma)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_\Gamma \leq Sim(\Gamma)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any Artin group of large type with $Sim(\Gamma)=3$ is exactly two.