The place of super edge-magic labelings among other classes of labelings

Abstract

A (p; q)-graph G is edge-magic if there exists a bijective function f : V (G)∪E(G) → {1; 2; : : : ; p + q} such that f(u) + f(v) + f(uv) = k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V (G)) = {1; 2; : : : ; p}. In this paper, we present some necessary conditions for a graph to be super edge-magic. By means of these, we study the super edge-magic properties of certain classes of graphs. We also exhibit the relationships between super edge-magic labelings and other well-studied classes of labelings. In particular, we prove that every super edge-magic (p; q)-graph is harmonious and sequential (for a tree or q ¿ p) as well as it is cordial, and sometimes graceful. Finally, we provide a closed formula for the number of super edge-magic graphs.

On Partitional Labelings of Graphs

“…The ladder is the graph P m × Q 1 , where P m is the path with m vertices. In [2], Figueroa-Centeno et al have shown that the ladder P 2m+1 × Q 1 has a super edge-magic labeling for every positive integer m, and it induces the partitional labeling of label. Thus, we have the following result.…”

On $H$-antimagicness of Cartesian product of graphs

“…However, it is worthwhile mentioning that a type of graph called strongly indexable had already been defined in [1] by Acharya and Hedge and it turns out that strongly indexable graphs are equivalent to super edge-magic graphs. For further information on (super) edge-antimagic total labelings, one can see [4,6,9,11,13,21].…”

Super edge-magic labeling of graphs: deficiency and maximality

“…A graph having a super edge-magic labeling is a super edgemagic graph. In [6], Figueroa-Centeno et al provided a useful characterization of a super edgemagic graph that we state in the following lemma. Lemma 1.1.…”

“…Lemma 1.1. [6] A graph G is super edge-magic if and only if there exists a bijective function f : V (G) → {1, 2, · · · , |V (G)|} such that the set S = {f (x) + f (y) : xy ∈ E(G)} is a set of |E(G)| consecutive integers.…”