On Super Edge-Magic Labelings of Unions of Star Graphs

Abstract

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 $f:V\cup E\to \{1,2,\dots ,p+q\}$ 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 $f(V)=\{1,2,\dots ,p\}$. 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 $r=3$.