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.