$E_A$-cordial labeling of graphs and its implications for $A$-antimagic labeling of trees

If $A$ is a finite Abelian group, then a labeling $f \colon E (G) \rightarrow A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex $u$ receives the label $\sum_{v\in N(u)}f (v)$, where $N(u)$ is an open neighborhood of the vertex $u$. A graph $G$ is $E_A$-cordial if there is an edge-labeling such that (1) the edge label classes differ in size by at most one and (2) the induced vertex label classes differ in size by at most one. Such a labeling is called $E_A$-cordial. In the literature, so far only $E_A$-cordial labeling in cyclic groups has been studied. The corresponding problem was studied by Kaplan, Lev and Roditty. Namely, they introduced $A^*$-antimagic labeling as a generalization of antimagic labeling \cite{ref_KapLevRod}. Simply saying, for a tree of order $|A|$ the $A^*$-antimagic labeling is such $E_A$-cordial labeling that the label $0$ is prohibited on the edges. In this paper, we give necessary and sufficient conditions for paths to be $E_A$-cordial for any cyclic $A$. We also show that the conjecture for $A^*$-antimagic labeling of trees posted in \cite{ref_KapLevRod} is not true.