Magic covering of chain of an arbitrary 2-connected simple graph

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 is a total labeling $f:V\cup E\to \{1,2,3,\dots ,|V|+|E\left|\right\}$ such that for each subgraph $H^{\prime }=(V^{\prime },E^{\prime })$ of $G$ isomorphic to $H$, ${\displaystyle {\sum }_{v\in V^{\prime }}f(v)+}{\displaystyle {\sum }_{e\in E^{\prime }}f(e)}$ is constant. When $f(V)=\{1,2,\dots ,|V\left|\right\}$, then $G$ is said to be $H$-supermagic. In this paper we show that a chain of any 2-connected simple graph $H$ is $H$-supermagic.