Supermagic Complete Graphs

Abstract

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.

On New Properties of Graphs with Magic Type Labeling

“…The results in regards to relation between closed distance magic labeling graph and its spectrum are described in study [10]. In this section, we continue posed to use sequential integers; he also called the edge labeling as supermagic [4]. Currently, there are various types of magic labelings of a graph.…”

Polytopes of Magic Labelings of Graphs and the Faces of the Birkhoff Polytope

“…Graphs with a magic labeling are also called magic graphs (see Figure 1 for an example of a magic labeling of the complete graph K 6 ). Magic labelings of graphs are studied in great detail by Stanley in [30,31], and Stewart in [32,33].…”

“…, g n = I}. The Cayley group digraph of G is a graphical representation of G: Every element g i of the group G corresponds to a vertex v i Figure 1: A magic labeling of the complete graph K 6 of magic sum 40 [33].…”

“…P K n is an n(n−3) 2 dimensional polytope with the description: See [33] for more aspects of magic labelings of complete graphs K n . The minimal Hilbert basis of the cone of magic labelings of the 1-skeleton of the tetrahedron is given in Figure 16.…”

Numbers of edges in supermagic graphs

“…It is easy to see that the classical concept of a magic square of n 2 boxes corresponds to the fact that the complete bipartite graph K n,n is supermagic for every positive integer n = 2 (see also [12]). Stewart [13] characterized supermagic complete graphs. In [6] supermagic regular complete multipartite graphs and supermagic cubes are characterized.…”