On Super Edge-Magic Graphs which are Weak Magic

Abstract

A \( (p,q) \) graph \( G \) is total edge-magic if there exists a bijection \( f: V \cup E \to \{1, 2, \ldots, p+q\} \) 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 \( 1 \leq \text{eMt}(G) – \text{emt}(G) \leq p \), and \textbf{weak magic} if \( \text{eMt}(G) – \text{emt}(G) > p \). A total edge-magic graph is called a super edge-magic if \( f(V(G)) = \{1, 2, \ldots, p\} \). The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.