  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.
