Relations for the Grothendieck groups of triangulated categories

A class of triangulated categories with a finiteness condition is singled out. These triangulated categories have Auslander–Reiten triangles. It is proved that the relations of the Grothendieck group of a triangulated category in this class are generated by all Auslander–Reiten triangles. Moreover, the Auslander–Reiten quivers of certain triangulated categories in this class are described in terms of Dynkin diagrams.