$H$-Supermagic Strength of Some 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 is a total labeling $f:V\cup E\rightarrow\{1,2,3,\ldots,|V|+|E|\}$ such that for each subgraph $H'=(V',E')$ of $G$ isomorphic to $H$, $s(f)=\sum_{v\in V'} f(v)+\sum_{e\in E'} f(e)$ is constant. When $f(V)=\{1,2,\ldots,|V|\}$, then $G$ is said to be $H$-supermagic. In this case, the $H$-supermagic strength of $G$ is defined as the minimum of all $s(f)$ where the minimum is taken over all $H$-supermagic labelings $f$ of $G$, and is denoted by $SM_H(G)$. In this paper we find the $C_k$-supermagic strength of k-polygonal snakes of any length and $H$-supermagic strength of a chain of an arbitrary 2-connected simple graph $H$. Also we make a conjecture regarding the $P_h$-supermagic strength of $P_n$ for $2 \leq h \leq n$.