Zᴋ-Magic Labeling of Path Union of Graphs

Abstract

For any non-trivial Abelian group $A$ under addition a graph $G$ is said to be $A$-\textit{magic} if there exists a labeling $f:E(G) \to A-\{0\}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = \sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-\textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$, the group of integers modulo $k$ and these graphs are referred as $k$-\textit{magic} graphs. In this paper we prove that the graphs such as path union of cycle, generalized Petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle and $n$-pan graph are $Z_k$-magic graphs.

Keywords

• a-magic labeling
• zk-magic labeling
• zk -magic graph
• generalized petersen graph
• shell
• wheel
• closed helm
• double wheel
• flower
• cylinder
• total graph of a path
• lotus inside a circle
• n-pan graph