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 \(\{1, 2, 3, \ldots, |V| + |E|\}\) such that \(\lambda(V) = \{1, 2, \ldots, |V|\}\) and \(\lambda(x) + \lambda(xy) + \lambda(y)\) is constant for every edge \(xy\).In this paper, given a positive integer \(k\) (\(k \geq 6\)), 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.