  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.
