If A Γ (W Γ ) is the Artin (Coxeter) group with defining graph Γ we denote by Sim(Γ) the number of vertices of the largest clique in Γ. We show that asdimA Γ ≤ Sim(Γ), if Sim(Γ) = 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 Γ ≤ Sim(Γ) 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(Γ) = 3 is exactly two.

Let W Γ be the Right-Angled Coxeter group with defining graph Γ. We show that the asymptotic dimension of W Γ is smaller than or equal to dim CC (Γ), the clique-connected dimension of the graph. As a corollary we show that W Γ is virtually free if and only if dim CC (Γ) = 1.Theorem 1.2. Let W Γ be the Right-Angled Coxeter group with defining graph Γ. ThenAs a corollary of theorem 1.1 we prove that:Interestingly, we can deduce whether W Γ is virtually free from the cliqueconnected dimension of its defining graph. In particular, we show the following:Proposition 1.4. Let W Γ be the Right-Angled Coxeter group with connected defining graph Γ. Then W Γ is virtually free if and only if dim CC (Γ) = 1.The paper is organized as follows. In subsection 3.1 we start with some basic definitions and some preliminary results that are used in the rest of the paper. Subsection 3.2 is containing some important lemmas, for example, we show that dim CC ( * ) is "monotone" in the following sense: if Γ ′ is a full-subgraphs of Γ, then dim CC (Γ ′ ) ≤ dim CC (Γ). In subsection 3.3, we prove that the clique-connected dimension is increasing in some cases. In subsection 3.4, we prove the main theorem of the paper. In the last subsection, we present some corollaries of the main theorem.Acknowledgments: I would like to thank Panos Papasoglu for his valuable advices during the development of this research work.