On Super Edge-Magic Graphs

Abstract

A graph $G=(V,E)$ is said to be super edge-magic if there exists a one-to-one correspondence $A$ from $V\cup E$ onto $\{1,2,3,\dots ,|V|+|E|\}$ such that $\lambda (V)=\{1,2,\dots ,|V|\}$ and $\lambda (x)+\lambda (xy)+\lambda (y)$ is constant for every edge $xy$.In this paper, given a positive integer $k$ ($k\ge 6$), we use the partitions of $k$ having three distinct parts to construct infinitely many super edge-magic graphs without isolated vertices with edge magic number $k$. Especially, we use this method to find graphs with the maximum number of edges among the super edge-magic graphs with $v$ vertices. In addition, we investigate whether or not some interesting families of graphs are super edge-magic.