Some new results on \(k\)-cordial labeling

Abstract

Hovey [11] called a graph \(G\) \(A\)-cordial, where \(A\) is an additive Abelian group, and \(f: V(G) \to A\) is a labeling of the vertices of \(G\) with elements of \(A\) such that when the edges of \(G\) are labeled by the induced labeling \(f: E(G) \to A\) by \(f^*(xy) = f(x) + f(y)\), then the number of vertices (resp. edges) labeled with \(\alpha\) and the number of vertices (resp. edges) labeled with \(\beta\) differ by at most one for all \(\alpha, \beta \in A\). When \(A = \mathbb{Z}_k\), we call a graph \(G\) \(k\)-cordial instead of \(\mathbb{Z}_k\)-cordial. In this paper, we give a sufficient condition for the join of two \(k\)-cordial graphs to be \(k\)-cordial and we give also a necessary condition for certain Eulerian graphs to be \(k\)-cordial when \(k\) is even, and finally we complete the characterization of the \(4\)-cordiality of the complete tripartite graph.