Susana C. López

BI-MAGIC AND OTHER GENERALIZATIONS OF SUPER EDGE-MAGIC LABELINGS

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.

ENUMERATING SUPER EDGE-MAGIC LABELINGS FOR THE UNION OF NONISOMORPHIC GRAPHS

A super edge-magic labeling of a graph G=(V,E) of order p and size q is a bijection f:V ∪E→{i}p+qi=1 such that: (1) f(u)+f(uv)+f(v)=k for all uv∈E; and (2) f(V )={i}pi=1. Furthermore, when G is a linear forest, the super edge-magic labeling of G is called strong if it has the extra property that if uv∈E(G) , u′,v′ ∈V (G) and dG (u,u′ )=dG (v,v′ )<+∞, then f(u)+f(v)=f(u′ )+f(v′ ). In this paper we introduce the concept of strong super edge-magic labeling of a graph G with respect to a linear forest F, and we study the super edge-magicness of an odd union of nonnecessarily isomorphic acyclic graphs. Furthermore, we find exponential lower bounds for the number of super edge-magic labelings of these unions. The case when G is not acyclic will be also considered.

The Polynomial Method

In Chap. 6 we discussed the existence of labelings by utilizing the ⊗ h -product of digraphs, which could be expressed algebraically as a generalization of voltage assignments, a classical technique used in topological graph theory. In this chapter, we introduce an algebraic method: Combinatorial Nullstellensatz.

Notation and Terminology

This chapter contains notation and terminology used in the book. The remaining concepts not found here and needed in the book will be defined in the corresponding chapter.

Super Edge Magic Labelings: First Type of Relations

The number of different types of graph labelings has become enormous during the last five decades. A good proof of that is the survey by Gallian [15]. It seems that researchers consider each labeling separately from the rest of labelings. The title of the second survey paper on graph labelings published by Gallian [14], “A guide to the graph labeling zoo” reflects very well this fact.

The ⊗-Product of Digraphs: Second Type of Relations

Since the beginning of graph labelings, researchers interested in this topic have dedicated their efforts mainly on finding techniques to prove the existence of particular families of graphs admitting some specific types of labeling. However, very few general techniques are known in order to create labelings of graphs.

Graceful Labelings: The Shifting Technique

Graceful labelings of graphs appeared in 1967 due to the relationship found with the problem of decompositions of graphs, in particular with the problem of decomposing complete graphs into copies of a given tree. Strong relations between graceful labelings and Golomb rulers (which are a different way to understand Sidon sets) were also found.

Distance labelings: a generalization of Langford sequences

A Langford sequence of order m and defect d can be identified with a labeling of the vertices of a path of order 2 m in which each label from d up to d  +  m  − 1 appears twice and in which the vertices that have been labeled with k are at distance k . In this paper, we introduce two generalizations of this labeling that are related to distances. The basic idea is to assign nonnegative integers to vertices in such a way that if n vertices ( n  > 1 ) have been labeled with k then they are mutually at distance k . We study these labelings for some well known families of graphs. We also study the existence of these labelings in general. Finally, given a sequence or a set of nonnegative integers, we study the existence of graphs that can be labeled according to this sequence or set.