Bi-Magic and Other Generalizations of Super Edge-Magic Labelings

Abstract

In this paper, we use the product ⊗ h in order to study super edge-magic labelings, bi-magic labelings and optimal k-equitable labelings. We establish, with the help of the product ⊗ h , new relations between super edge-magic labelings and optimal k-equitable labelings and between super edge-magic labelings and edge bi-magic labelings. We also introduce new families of graphs that are inspired by the family of generalized Petersen graphs. The concepts of super bi-magic and r -magic labelings are also introduced and discussed, and open problems are proposed for future research.2010 Mathematics subject classification: primary 05C78.

A new labeling construction from the⊗h-product

“…In all the results involving the ⊗ h -product, since the very beginning, it seems to be a constant to use super edge-magic labeled graphs as the second factor of the product, or at least graphs that in a way or another come from super edge-magic graphs [10,15,18]. The power of this section lies in the fact that it allows us to use other types of labeled graphs as a second factor of the product and this allows to refresh the ways of attacking old famous problems in the subject of graph labelings as we will in the next lines.…”

“…Many relations among labelings have been established using the ⊗ h -product and some particular families of graphs, namely S p and S k p (see for instance, [10,15,18,19]). The family S p contains all super edge-magic 1-regular labeled digraphs of order p where each vertex takes the name of the label that has been assigned to it.…”

Langford sequences and a product of digraphs

“…Theorem 2.5. [18] Let D be an (optimal) k-equitable digraph and let h : E(D) → RS n be any function. Then D ⊗ h RS n is (optimal) k-equitable.…”

The Jumping Knight and Other (Super) Edge-Magic Constructions

“…Super edge-magic labelings are of importance among graph labelings due to the great amount of relations that they have with other labelings (see [7,11,13,16,17]).…”