Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

Now streamlined from previous editions, a new author team brings in a fresh look to this classic textbook and at how graph theory courses have evolved. When the first edition of this precursor text was published, there were few undergraduate courses offered. The text assisted in the establishment of the undergraduate course, while also offering enough coverage for a graduate course. Graph theory is not a seminal course in all combinatorics programs taught in universities and colleges throughout the world. This text has remained among the top three best-sellers. The book is famous for the quality of the writing and presentation. We have two best-sellers for his course, including Gross, etal.'s Graph Theory and Its Applications. This one-two punch in this course means we can go against any and all texts and compete successfully.

With Chromatic Graph Theory, Second Edition, the authors present various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. Readers will see that the authors accomplished the primary goal of this textbook, which is to introduce graph theory with a coloring theme and to look at graph colorings in various ways. The textbook also covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings. Features of the Second Edition: The book can be used for a first course in graph theory as well as a graduate course The primary topic in the book is graph coloring The book begins with an introduction to graph theory so assumes no previous course The authors are the most widely-published team on graph theory Many new examples and exercises enhance the new edition

Graphs & Digraphs, Seventh Edition masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. This classic text, widely popular among students and instructors alike for decades, is thoroughly streamlined in this new, seventh edition, to present a text consistent with contemporary expectations.

Changes and updates to this edition include:

A rewrite of four chapters from the ground up

Streamlining by over a third for efficient, comprehensive coverage of graph theory

Flexible structure with foundational Chapters 1–6 and customizable topics in Chapters 7–11

Incorporation of the latest developments in fundamental graph theory

Statements of recent groundbreaking discoveries, even if proofs are beyond scope

Completely reorganized chapters on traversability, connectivity, coloring, and extremal graph theory to reflect recent developments

The text remains the consummate choice for an advanced undergraduate level or introductory graduate-level course exploring the subject’s fascinating history, while covering a host of interesting problems and diverse applications. Our major objective is to introduce and treat graph theory as the beautiful area of mathematics we have always found it to be. We have striven to produce a reader-friendly, carefully written book that emphasizes the mathematical theory of graphs, in all their forms. While a certain amount of mathematical maturity, including a solid understanding of proof, is required to appreciate the material, with a small number of exceptions this is the only pre-requisite.

In addition, owing to the exhilarating pace of progress in the field, there have been countless developments in fundamental graph theory ever since the previous edition, and many of these discoveries have been incorporated into the book. Of course, some of the proofs of these results are beyond the scope of the book, in which cases we have only included their statements. In other cases, however, these new results have led us to completely reorganize our presentation. Two examples are the chapters on coloring and extremal graph theory.

Table of Contents

1 Graphs

1.1 Fundamentals

1.2 Isomorphism

1.3 Families of graphs

1.4 Operations on graphs

1.5 Degree sequences

1.6 Path and cycles

1.7 Connected graphs and distance

1.8 Trees and forests

1.9 Multigraphs and pseudographs

2 Digraphs

2.1 Fundamentals

2.2 Strongly connected digraphs

2.3 Tournaments

2.4 Score sequences

3 Traversability

3.1 Eulerian graphs and digraphs

3.2 Hamiltonian graphs

3.3 Hamiltonian digraphs

3.4 Highly hamiltonian graphs

3.5 Graph powers

4 Connectivity

4.1 Cut-vertices, bridges, and blocks

4.2 Vertex connectivity

4.3 Edge-connectivity

4.4 Menger's theorem

5 Planarity

5.1 Euler's formula

5.2 Characterizations of planarity

5.3 Hamiltonian planar graphs

5.4 The crossing number of a graph

6 Coloring

6.1 Vertex coloring

6.2 Edge coloring

6.3 Critical and perfect graphs

6.4 Maps and planar graphs

7 Flows

7.1 Networks

7.2 Max-flow min-cut theorem

7.3 Menger's theorems for digraphs

7.4 A connection to coloring

8 Factors and covers

8.1 Matchings and 1-factors

8.2 Independence and covers

8.3 Domination

8.4 Factorizations and decompositions

8.5 Labelings of graphs

9 Extremal graph theory

9.1 Avoiding a complete graph

9.2 Containing cycles and trees

9.3 Ramsey theory

9.4 Cages and Moore graphs

10 Embeddings

10.1 The genus of a graph

10.2 2-Cell embeddings of graphs

10.3 The maximum genus of a graph

10.4 The graph minor theorem

11 Graphs and algebra

11.1 Graphs and matrices

11.2 The automorphism group

11.3 Cayley color graphs

11.4 The reconstruction problem

With Chromatic Graph Theory, Second Edition, the authors present various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. Readers will see that the authors accomplished the primary goal of this textbook, which is to introduce graph theory with a coloring theme and to look at graph colorings in various ways. The textbook also covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings. Features of the Second Edition: The book can be used for a first course in graph theory as well as a graduate course The primary topic in the book is graph coloring The book begins with an introduction to graph theory so assumes no previous course The authors are the most widely-published team on graph theory Many new examples and exercises enhance the new edition

Die Theorie der regularen Graphen (The Theory of Regular Graphs), written by the Danish Mathematician Julius Petersen in 1891, is often considered the first strictly theoretical paper dealing with graphs. In the 130 years since then, regular graphs have been a common and popular area of study. While regular graphs are typically considered to be graphs whose vertices all have the same degree, a more general interpretation is that of graphs possessing some common characteristic throughout their structure. During the past several decades, however, there has been some increased interest in investigating graphs possessing a property that is, in a sense, opposite to regularity. It is this topic with which this book deals, giving rise to a study of what might be called irregularity in graphs. Here, various irregularity concepts dealing with several topics in graph theory are described, such as degrees of vertices, graph labelings, weightings, colorings, graph structures, Eulerian and Hamiltonian properties, graph decompositions, and Ramsey-type problems.

This book is in honor of the 80th birthday of Stephen Hedetniemi. It describes advanced material in graph theory in the areas of domination, coloring, spanning cycles and circuits, and distance that grew out of research topics investigated by Stephen Hedetniemi. The purpose of this book is to provide background and principal results on these topics, along with same related problems and conjectures, for researchers in these areas. The most important features deal with material, results, and problems that researchers may not be aware of but may find of interest. Each chapter contains results, methods and information that will give readers the necessary background to investigate each topic in more detail.

Graph theory goes back several centuries and revolves around the study of graphs--mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics--and some of its most famous problems. The Fascinating World of Graph Theory explores the questions and puzzles that have been studied, and often solved, through graph theory. This book looks at graph theory's development and the vibrant individuals responsible for the field's growth. Introducing fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, and each chapter contains math exercises for readers to savor. An eye-opening journey into the world of graphs, The Fascinating World of Graph Theory offers exciting problem-solving possibilities for mathematics and beyond.

Graphs & Digraphs masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. Fully updated and thoughtfully reorganized to make reading and locating material easier for instructors and students, the Sixth Edition of this bestselling, classroom-tested text: Adds more than 160 new exercises Presents many new concepts, theorems, and examples Includes recent major contributions to long-standing conjectures such as the Hamiltonian Factorization Conjecture, 1-Factorization Conjecture, and Alspach's Conjecture on graph decompositions Supplies a proof of the perfect graph theorem Features a revised chapter on the probabilistic method in graph theory with many results integrated throughout the text At the end of the book are indices and lists of mathematicians' names, terms, symbols, and useful references. There is also a section giving hints and solutions to all odd-numbered exercises. A complete solutions manual is available with qualifying course adoption. Graphs & Digraphs, Sixth Edition remains the consummate text for an advanced undergraduate level or introductory graduate level course or two-semester sequence on graph theory, exploring the subject's fascinating history while covering a host of interesting problems and diverse applications.

This book depicts graph labelings that have led to thought-provoking problems and conjectures. Problems and conjectures in graceful labelings, harmonious labelings, prime labelings, additive labelings, and zonal labelings are introduced with fundamentals, examples, and illustrations. A new labeling with a connection to the four color theorem is described to aid mathematicians to initiate new methods and techniques to study classical coloring problems from a new perspective. Researchers and graduate students interested in graph labelings will find the concepts and problems featured in this book valuable for finding new areas of research.

For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number theory, combinatorics, and calculus. The exercises receive consistent praise from users for their thoughtfulness and creativity. They help students progress from understanding and analyzing proofs and techniques to producing well-constructed proofs independently. This book is also an excellent reference for students to use in future courses when writing or reading proofs. 0134746759 / 9780134746753 Chartrand/Polimeni/Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, 4/e

Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics. New to the Fifth Edition * New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings * New examples, figures, and applications to illustrate concepts and theorems * Expanded historical discussions of well-known mathematicians and problems * More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book * Reorganization of sections into subsections to make the material easier to read * Bolded definitions of terms, making them easier to locate Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subject's fascinating history and presents a host of interesting problems and diverse applications.