On Super Edge-Magic Labelings of Unions of Star Graphs

Let \( G = (V,E) \) be a graph with \( |V| = p \) and \( |E| = q \). The graph \( G \) is total edge-magic if there exists a bijection \( f : V \cup E \to \{1,2,\ldots,p+q\} \) such that for all \( e = (u,v) \in E \), \( f(u) + f(e) + f(v) \) is constant throughout the graph. A total edge-magic graph is called super edge-magic if \( f(V) = \{1,2,\ldots,p\} \). Lee and Kong conjectured that for any odd positive integer \( r \), the union of any \( r \) star graphs is super edge-magic. In this paper, we supply substantial new evidence to support this conjecture for the case \( r = 3 \).