Super Edge-Magic Labeling of 5-Uniform and 6-Uniform Hypergraphs Generated by Arbitrary Simple Graphs

Authawich Narissayaporn, Ratinan Boonklurb, Sirirat Singhun

Abstract

The super edge-magic (SEM) labeling on hypergraphs is the extension of the SEM labeling on graphs. For a hypergraph $H$ with vertex set $V_H$ and hyperedge set $E_H$, we call a bijective mapping $f:V_H cup E_H to {1,2,3,\ldots,|V_H|+|E_H|\}$ as an SEM labeling of $H$ if and only if (i) there is an integer $\Lambda$ such that for every $e\in E_H$, $f(e)+\sum_{v\in e} f(v)=\Lambda$ and (ii) $f(V_H)=\{1,2,3,\ldots,|V_H|\}$. In this article, we define $5$-uniform $H^{(5)}(G)$ and $6$-uniform $H^{(6)}(G)$ hypergraphs from an arbitrary simple graph $G$ and show that $H^{(5)}(G)$ is always an SEM hypergraph. However, if $G$ has odd number of edges, then $H^{(6)}(G)$ is an SEM hypergraph. Unfortunately, if $G$ has even number of edges, the SEM labeling for $H^{(6)}(G)$ depends on the structure of the hypergraph. Thus, an example of SEM labeling of $H^{(6)}(nC_4)$, which has even number of edges, is given. Finally, if $H$ is a $k$-uniform SEM hypergraph, then we can show that $H'$, obtained from $H$ by adding more vertices, is $(k+2m)$-uniform SEM hypergraph.