(2, 3)-cordial trees and paths (Q6616811)

In 2022, \textit{L. B. Beasley} [J. Comb. Math. Comb. Comput. 117, 1--23 (2022; \url{doi:10.61091/jcmcc121-06})] introduced (2,3)-cordial labelings of directed graphs. He posed the following two conjectures 1. Every orientation of every path is (2,3)-cordial except for a path with four vertices. 2. Every tree of maximum degree 3 is (2,3)-orientable. The authors proved that both conjectures are not true. They proved the existence of an orientation of the ten path which is not (2,3)-cordial and so the first Conjecture fails. They also exhibited a tree of maximum degree 3 that is not (2,3)-orientable to disprove the second conjecture. They also proved that the Petersen graph is not (2,3)-orientable.\N\NFor the entire collection see [Zbl 1540.05004].