Cyclical Orientations Of Graphs

Cluster algebras appeared just several years ago. It has been shown that cluster algebras are related to various subjects. S. Fomin and A. Zelevinsky defined and classified the cluster algebra of finite type. By a result of M. Barot,C. Geiss and A. Zelevinsky, finite type for a cluster algebra depends on the cyclical orientation of the corresponding graph. They gave a method to decide whether a graph can be cyclically oriented. Their method is either to consider the linear order of the edges such that different chordless cycles have different maximal edges, or to check the number relation among the amount of vertices, edges, cycles and connected components. In this paper, we give another method to decide whether a given graph can be cyclically oriented. Our method is to use the chains between two vertices.