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2017 | Buch

Notation and Terminology Kapitel lesen Erstes Kapitel lesen

Graceful, Harmonious and Magic Type Labelings

Relations and Techniques

verfasst von: Susana C. López, Francesc A. Muntaner-Batle

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Aimed toward upper undergraduate and graduate students in mathematics, this book examines the foremost forms of graph labelings including magic, harmonious, and graceful labelings. An overview of basic graph theory concepts and notation is provided along with the origins of graph labeling. Common methods and techniques are presented introducing readers to links between graph labels. A variety of useful techniques are presented to analyze and understand properties of graph labelings. The classical results integrated with new techniques, complete proofs, numerous exercises, and a variety of open problems, will provide readers with a solid understanding of graph labelings.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Notation and Terminology
Abstract
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.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 2. Graphs Labelings
Abstract
By a labeling of a graph, also known as a valuation of a graph, we mean a map that carries graph elements onto numbers (usually the positive or nonnegative integers) called labels that meet some properties depending on the type of labeling that we are considering. The most common choices for the domain are the set of vertices alone (vertex labelings), or edges alone (edge labelings), or the set of edges and vertices together (total labelings) (see [12]). Other domains are also possible, but they will not be discussed in this book.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 3. Super Edge Magic Labelings: First Type of Relations
Abstract
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.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 4. Harmonious Labelings
Abstract
Harmonious labelings are very important in the literature, since many authors have devoted their efforts to better understanding them. Thus, we believe that it is worth the while to dedicate special attention to them. However many of the results that involve harmonious labelings can be obtained using similar ideas to the ones used for super edge-magic labelings as well as using the relations that appear in Sects. 3.​1 and 6.​3 Therefore, we will dedicate only a short chapter to this important labeling. We start answering the question of which complete graphs are harmonious. Later we will study a few families of harmonious graphs as well as general properties of these graphs. We will conclude this chapter providing an asymptotic answer to the question of how many graphs are harmonious.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 5. Graceful Labelings: The Shifting Technique
Abstract
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.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 6. The ⊗-Product of Digraphs: Second Type of Relations
Abstract
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.
Susana C. López, Francesc A. Muntaner-Batle
Chapter 7. The Polynomial Method
Abstract
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.
Susana C. López, Francesc A. Muntaner-Batle
Metadaten
Titel
Graceful, Harmonious and Magic Type Labelings
verfasst von
Susana C. López
Francesc A. Muntaner-Batle
Copyright-Jahr
2017
Electronic ISBN
978-3-319-52657-7
Print ISBN
978-3-319-52656-0
DOI
https://doi.org/10.1007/978-3-319-52657-7

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