On Super Edge-Magic Graphs which are Weak Magic

Abstract

A $(p,q)$ graph $G$ is total edge-magic if there exists a bijection $f:V\cup E\to \{1,2,\dots ,p+q\}$ such that for each $e=(u,v)\in E$, we have $f(u)+f(e)+f(v)$ as a constant. For a graph $G$, denote $M(G)$ the set of all total edge-magic labelings. The magic strength of $G$ is the minimum of all constants among all labelings in $M(G)$, denoted by $\text{emt}(G)$. The maximum of all constants among $M(G)$ is called the maximum magic strength of $G$ and denoted by $\text{eMt}(G)$.

Hegde and Shetty classify a magic graph as strong if $\text{emt}(G)=\text{eMt}(G)$, ideal magic if $1\le \text{eMt}(G)–\text{emt}(G)\le p$, and $\mathbf{\text{weak magic}}$ if $\text{eMt}(G)–\text{emt}(G)>p$. A total edge-magic graph is called a super edge-magic if $f(V(G))=\{1,2,\dots ,p\}$. The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.