On the super edge-magicness of graphs of equal order and size

TL;DR

This paper investigates the super edge-magic properties of graphs with equal order and size, providing new results and negative findings that challenge common assumptions in the field of graph labelings and combinatorics.

Contribution

It offers novel insights into super edge-magic graphs of equal order and size, including unexpected negative results that defy typical parity and size conditions.

Findings

Negative results on super edge-magicness under certain conditions

Identification of cases where previous size and parity assumptions do not hold

Potential applications in combinatorics and design theory

Abstract

The super edge-magicness of graphs of equal order and size has been shown to be important since such graphs can be used as seeds to answer many questions related to (super) edge-magic labelings and other types of well studied labelings, as for instance harmonious labelings. Also other questions related to the area of combinatorics can be attacked and understood from the point of view of super edge-magic graphs of equal order and size. For instance, the design of Steiner triple systems, the study of the set of dual shuffle primes and the Jacobsthal numbers. In this paper, we study the super edge-magic properties of some types of super edge-magic graphs of equal order and size, with the hope that they can be used later in the study of other related questions. The negative results found in last section are specially interesting since these kind of results are not common in the literature. Furthermore, the few results found in this direction usually meet one of the following reasons: too many vertices compared with the number of edges; too many edges compared with the number of vertices; or parity conditions. In this case, all previous reasons fail in our results.