Graph Theory: An Introductory Course

ISBN-13: 9781461299691 / Angielski / Miękka / 2012 / 180 str.

Bela Bollobas

Opis

From the reviews: "Bela Bollobas introductory course on graph theory deserves to be considered as a watershed in the development of this theory as a serious academic subject. ... The book has chapters on electrical networks, flows, connectivity and matchings, extremal problems, colouring, Ramsey theory, random graphs, and graphs and groups. Each chapter starts at a measured and gentle pace. Classical results are proved and new insight is provided, with the examples at the end of each chapter fully supplementing the text... Even so this allows an introduction not only to some of the deeper results but, more vitally, provides outlines of, and firm insights into, their proofs. Thus in an elementary text book, we gain an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject. It is this aspect of the book which should guarantee it a permanent place in the literature." #Bulletin of the London Mathematical Society#1

Spis treści

I Fundamentals.- 1. Definitions.- 2. Paths, Cycles and Trees.- 3. Hamilton Cycles and Euler Circuits.- 4. Planar Graphs.- 5. An Application of Euler Trails to Algebra.- Exercises.- Notes.- II Electrical Networks.- 1. Graphs and Electrical Networks.- 2. Squaring the Square.- 3. Vector Spaces and Matrices Associated with Graphs.- Exercises.- Notes.- III Flows, Connectivity and Matching.- 1. Flows in Directed Graphs.- 2. Connectivity and Menger’s Theorem.- 3. Matching.- 4. Tutte’s 1-Factor Theorem.- Exercises.- Notes.- IV Extremal Problems.- 1. Paths and Cycles.- 2. Complete Subgraphs.- 3. Hamilton Paths and Cycles.- 4. The Structure of Graphs.- Exercises.- Notes.- V Colouring.- 1. Vertex Colouring.- 2. Edge Colouring.- 3. Graphs on Surfaces.- Exercises.- Notes.- VI Ramsey Theory.- 1. The Fundamental Ramsey Theorems.- 2. Monochromatic Subgraphs.- 3. Ramsey Theorems in Algebra and Geometry.- 4. Subsequences.- Exercises.- Notes.- VII Random Graphs.- 1. Complete Subgraphs and Ramsey Numbers—The Use of the Expectation.- 2. Girth and Chromatic Number—Altering a Random Graph.- 3. Simple Properties of Almost All Graphs—The Basic Use of Probability.- 4. Almost Determined Variables—The Use of the Variance.- 5. Hamilton Cycles—The Use of Graph Theoretic Tools.- Exercises.- Notes.- VIII Graphs and Groups.- 1. Cayley and Schreier Diagrams.- 2. Applications of the Adjacency Matrix.- 3. Enumeration and Pólya’s Theorem.- Exercises.- Notes.- Index of Symbols.