Vertex-Magic Labelings for Complete 3-Uniform Hypergraphs with 4n Vertices where n is Odd

Authawich Narissayaporn, Ratinan Boonklurb, Sirirat Singhun

Abstract

Let $H$ be a hypergraph with the  vertex set $V_H$ and the hyperedge set $E_H$. For $v\in V_H$,  denote $\nbhd(v)=\{e\in E_H\ |\ v\in e\}$. We generalize the definition of vertex-magic labeling in graph into the definition of vertex-magic labeling in hypergraph as follow. A vertex-magic labeling of $H$ is a bijective mapping $f:V_H\cup E_H \to \{1,2,3,\ldots,|V_H|+|E_H|\}$ with a vertex-magic constant $\Lambda$ such that for every $v\in V_H,f(v)+\sum_{e\in\nbhd(v)}f(e)=\Lambda$. This paper constructs some magic rectangle sets and applies them to determine a vertex-magic labeling for a complete $3$-uniform hypergraph with $4n$ vertices where $n\in\{1,3,5,\ldots\}$.