On some graph-cordial Abelian groups

Hovey introduced $A$-cordial labelings as a generalization of cordial and harmonious \cite{Hovey}. If $A$ is an Abelian group, then labeling $f \colon V (G) \rightarrow A$ the vertices some graph $G$ induces edge on $G$; $uv$ receives label (u) + f (v)$. A if there vertex-labeling such that (1) vertex classes differ in size by at most one (2) induced one. Patrias Pechenik studied larger class finite abelian groups all path graphs are $A$-cordial. They posed conjecture but finitely many paths for any group $A$. In this paper we solve conjecture. Moreover show cycle odd order.