(2, 3)-Cordial Trees and Paths

Recently L. B. Beasley introduced (2, 3)-cordial labelings of directed graphs in [1]. He conjectured that every orientation of a path of length at least five is (2, 3)-cordial, and that every tree of max degree \(n =3\) has a cordial orientation. In this paper we formally define (2, 3)-cordiality from the viewpoint of quasigroup cordiality. We show both conjectures to be false, discuss the (2, 3)-cordiality of orientations of the Petersen graph, and establish an upper bound for the number of edges a graph can have and still be (2, 3)-orientable.