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

If A is a finite Abelian group, then a labeling $f:E\left(G\right)\to A$ of the edges of some graph G induces a vertex labeling on G; the vertex u receives the label ${\sum }_{v\in N\left(u\right)}f\left(uv\right)$, where $N\left(u\right)$ 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.

Kaplan, Lev, and Roditty studied the corresponding problem. Namely, they introduced $A^{⁎}$-antimagic labeling as a generalization of antimagic labeling [11]. Simply saying, for a tree of order $\left|A\right|$ 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 [11] is not true.