On $E_k$ — Cordial Labeling

Abstract

Cahit and Yilmaz $[15]$ called a graph $G$ is $E_k$-cordial if it is possible to label its edges with numbers from the set $\{0,1,\dots ,k-1\}$ in such a way that, at each vertex $V$ of $G$, the sum modulo $k$ of the labels on the edges incident with $V$ satisfies the inequalities $|m_{(}i)–m_{(}j)|\le 1$ and $|n_{(}i)–n_{(}j)|\le 1$, where $m_{(}s)$ and $n_{(}t)$ are, respectively, the number of edges labeled with $s$ and the number of vertices labeled with $t$. In this paper, we give a necessary condition for a graph to be $E_k$-cordial for certain $k$. We also give some new families of $E_k$-cordial graphs and we prove Lee’s conjecture about the edge-gracefulness of the disjoint union of two cycles.