On some graph-cordial Abelian groups

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