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 \(\{0, 1, \ldots, k-1\}\) 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 \(|m_(i) – m_(j)| \leq 1\) and \(|n_(i) – n_(j)| \leq 1\), 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.