  Recently L. B. Beasley introduced $(2,3)$-cordial labelings of directed
graphs in [1]. He made two conjectures which we resolve in this article. 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. We
show these two conjectures to be false. We also 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 $(2,3)$ cordial. An application
of $(2,3)$ cordial labelings is also presented.

Combinatorics