Star magic labeling of some special graphs

Abstract

The Klein 4-group, denoted by $V_4$ is an abelian group of order 4. It has elements $V_4 =\{0, a, b, c\},$
with $a+a=b+b=c+c=0$ and $a+b=c,b+c=a,c+a=b.$ A graph $G = (V (G),E(G)),$ with
vertex set $V(G)$ and edge set $E(G),$ is said to be $star$ magic if there exists a labeling $f:V(G)\rightarrow V_4\backslash\{0\}$ such that the induced mapping $V_f^+:V(G)\rightarrow V_4$ defined by $V_f^+(v)=\sum_{u\in N(v)} f^*(uv),$ where $f^*(uv)=f(u)+f(v)$ 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 $p-star$ magic labeling of $G$ and $G$ is said to be a $p-star$ magic graph. If this constant is $0,$ then we say that $f$ is a $0-star$ magic labeling of $G$ and $G$ is said to be a $0-star$ magic graph.