Induced V_4-magic labeling of some subdivision graphs

Abstract

Let $V_4=\{ 0,a,b,c \}$ be the Klein-4-group with identity element $0$ 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 labeling of $f,$ defined by $f^* (v_1v_2)=f(v_1)+f(v_2)$ for all $v_1v_2\in E .$ Then $f^*$ again induces a labeling say $f^{**}: V \rightarrow V_4$ defined by $\displaystyle{f^{**}(v)=\sum_{vv_1\in E}} f^*(vv_1).$ A graph $G=(V ,E )$ is said to be an Induced $V_4$-Magic Graph 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 of $G$ and a graph which has no such induced magic labeling is called a non-induced magic graph. In this paper, we discuss some Induced $V_4$ magic labeling of some sub division graphs.