Neighbourhood \(V_4\)-magic labeling of some subdivision graphs

Abstract

Let \(V_4=\{0, a, b, c\}\) be the Klein-4-group with identity element 0. A graph \(G=(V(G), E(G))\), with vertex set \(V(G)\) and edge set \(E(G)\), is said to be Neighbourhood \(V_4\)-magic if there exists a labeling \(f: V(G) \rightarrow V_4 \backslash\{0\}\) such that the sum \(N_f^{+}(v)=\sum_{u \in N(v)} f(u)\) is a constant map. If this constant is \(p\), where \(p\) is any non zero element in \(V_4\), then we say that \(f\) is a \(p\)-neighbourhood \(V_4\)-magic labeling of \(G\) and \(G\) is said to be a \(p\)-neighbourhood \(V_4\)-magic graph. If this constant is 0 ,then we say that \(f\) is a 0 -neighbourhood \(V_4\)-magic labeling of \(G\) and \(G\) is said to be a 0 -neighbourhood \(V_4\)-magic graph.