Most wheel related graphs are not vertex magic

Abstract

Suppose $G$ is a finite graph with vertex-set $V(G)$ and edge-set $E(G)$. A one-to-one map $\lambda$ from $V(G)\cup E(G)$ onto the integers $1,2,3, \dots, |V(G)|+|E(G)|$ is called a {\it vertex-magic total labeling}, if there exists a constant $h$ so that for every vertex $x$, $$ \lambda(x) + \sum \lambda (xy) =h $$ where the sum is taken over all vertices $y$ adjacent to $x$. The constant $h$ is called the {\it magic constant} for $\lambda$. A graph with a vertex-magic total labeling will be called {\it vertex-magic}. In this paper, we consider the vertex-magic total labeling of wheel related graphs such as Jahangir graphs, helms, webs, flower graphs and sunflower graphs.