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