Induced \(V_4\)- magic labeling of cycle related graphs

Abstract

Let $V_4=\{0, a, b, c\}$ be the Klein-4-group with identity element 0 and $G=(V, E)$ be a graph. Let $f: V \rightarrow V_4$ be a vertex labeling and $f^*: E \rightarrow V_4$ be the induced edge labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a vertex labeling say $f^{* *}: V \rightarrow V_4$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $V_4$-Magic Graph (IMG) if there exists a non zero labeling $f: V \rightarrow V_4$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $V_4$-Magic labeling (IML) of $G$ and a graph which has no such induced magic labeling is called a non-induced magic graph. In this paper, we discuss Induced $V_4-$ magic labeling of some cycle related graphs.