Edge induced \(V_4\)− magic labeling of graphs

Abstract

Let $V_4=\{0, a, b, c\}$ be the Klein-4-group with identity element 0 and $G=(V(G), E(G))$ be the graph with vertex set $V(G)$ and edge set $E(G)$. Let $f: E(G) \rightarrow V_4 \backslash\{0\}$ be an edge labeling and $f^{+}: V(G) \rightarrow V_4$ denote the induced vertex labeling of $f$ defined by $f^{+}(u)=\sum_{u v \in E(G)} f(u v)$ for all $u \in\left(V(G)\right.$. Then $f^{+}$again induces an edge labeling $f^{++}: E(G) \rightarrow V_4$ defined by $f^{++}(u v)=f^{+}(u)+f^{+}(v)$. Then a graph $G=(V(G), E(G))$ is said to be an edge induced $V_4$-Magic graph if $f^{++}$is constant function. The function $f$, so obtained is called an edge induced $V_4$-Magic labeling of $G$. In this paper we discuss edge induced $V_4$ magic labeling of some graphs.