Strong \(\mathbb{Z}_{4p}\)- Magic Labeling

Abstract

For any non-trivial abelian group \(A\) under addition, a graph \(G\) is said to be strong \(A\)-magic if there exists a labeling \(f\) of the edges of \(G\) with non-zero elements of \(A\) such that the vertex labeling \(f^+\) defined as \(f^+(v) = \sum f(uv)\) taken over all edges \(uv\) incident at \(v\) is a constant, and the constant is same for all possible values of \(|V(G)|\). A graph is said to be strong \(A\)-magic if it admits strong \(A\)-magic labeling. In this paper, we consider \((\mathbb{Z}_4, +)\) as an abelian group and we prove strong \(\mathbb{Z}_4\)-magic labeling for various graphs and generalize strong \(\mathbb{Z}_{4p}\)-magic labeling for those graphs. The graphs which admit strong \(\mathbb{Z}_{4p}\)-magic labeling are called as strong \(\mathbb{Z}_{4p}\)-magic graphs.