Labeled four cycles and the \(K(\pi ,1)\)-conjecture for Artin groups

We show that for a large class of Artin groups with Dynkin diagrams being a tree, the $K(\pi ,1)$ $K(\pi ,1)$ -conjecture holds. We also establish the $K(\pi ,1)$ $K(\pi ,1)$ -conjecture for another class of Artin groups whose Dynkin diagrams contain a cycle, which applies to some Artin groups whose Dynkin diagrams are of hyperbolic type. This is based on a new approach to the $K(\pi ,1)$ $K(\pi ,1)$ -conjecture for Artin groups.