Some $C_4$- supermagic Graphs

Abstract

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to a subgraph of $G$ isomorphic to $H$. We say that $G$ is $H$-magic if there exists a total labeling $f:V\cup E\to \{1,2,\dots ,|V|+|E|+1\}$ such that for each subgraph $H^{\prime }=(V^{\prime },E”)$ of $G$ isomorphic to $H$,
$\sum _{v\in V^{\prime }}f(v)+\sum _{e\in E”}f(e)$
is constant.

When $f(V)=\{1,2,\dots ,|V|\}$, then $G$ is said to be $H$-supermagic.

In this paper, we show that all prism graphs $C_n\times P_m$, except for $n=4$, the ladder graph $P_3\times P_n$, and the grid $P_3\times P_n$, are $C_4$-supermagic.