Graph theory has experienced a tremendous growth during the 20th century. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. This book aims to provide a solid background in the basic topics of graph theory. It covers Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem on the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier's proof of Kuratowski's theorem on planar graphs, the proof of the nonhamiltonicity of the Tutte graph on 46 vertices and a concrete application of triangulated graphs. The book does not presuppose deep knowledge of any branch of mathematics, but requires only the basics of mathematics. It can be used in an advanced undergraduate course or a beginning graduate course in graph theory.
Reihe
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
Dateigröße
ISBN-13
978-1-4419-8505-7 (9781441985057)
DOI
10.1007/978-1-4419-8505-7
Schweitzer Klassifikation
I Basic Results.- 1.0 Introduction.- 1.1 Basic Concepts.- 1.2 Subgraphs.- 1.3 Degrees of Vertices.- 1.4 Paths and Connectedness.- 1.5 Automorphism of a Simple Graph.- 1.6 Line Graphs.- 1.7 Operations on Graphs.- 1.8 An Application to Chemistry.- 1.9 Miscellaneous Exercises.- Notes.- II Directed Graphs.- 2.0 Introduction.- 2.1 Basic Concepts.- 2.2 Tournaments.- 2.3 K-Partite Tournaments.- Notes.- III Connectivity.- 3.0 Introduction.- 3.1 Vertex Cuts and Edge Cuts.- 3.2 Connectivity and Edge-Connectivity.- 3.3 Blocks.- 3.4 Edge-Connectivity of a Graph.- 3.5 Menger's Theorem.- 3.6 Exercises.- Notes.- IV Trees.- 4.0 Introduction.- 4.1 Definition, Characterization, and Simple Properties.- 4.2 Centers and Centroids.- 4.3 Counting the Number of Spanning Trees.- 4.4 4.4 Cayley's Formula.- 4.5 Helly Property.- 4.6 Exercises.- Notes.- V Independent Sets and Matchings.- 5.0 Introduction.- 5.1 Vertex Independent Sets and Vertex Coverings.- 5.2 Edge-Independent Sets.- 5.3 Matchings and Factors.- 5.4 Matchings in Bipartite Graphs.- 5.5 * Perfect Matchings and the Tutte Matrix.- Notes.- VI Eulerian and Hamiltonian Graphs.- 6.0 Introduction.- 6.1 Eulerian Graphs.- 6.2 Hamiltonian Graphs.- 6.3 * Pancyclic Graphs.- 6.4 Hamilton Cycles in Line Graphs.- 6.5 2-Factorable Graphs.- 6.6 Exercises.- Notes.- VII Graph Colorings.- 7.0 Introduction.- 7.1 Vertex Colorings.- 7.2 Critical Graphs.- 7.3 Triangle-Free Graphs.- 7.4 Edge Colorings of Graphs.- 7.5 Snarks.- 7.6 Kirkman's Schoolgirls Problem.- 7.7 Chromatic Polynomials.- Notes.- VIII Planarity.- 8.0 Introduction.- 8.1 Planar and Nonplanar Graphs.- 8.2 Euler Formula and Its Consequences.- 8.3 K5 and K3,3 are Nonplanar Graphs.- 8.4 Dual of a Plane Graph.- 8.5 The Four-Color Theorem and the Heawood Five-Color Theorem.- 8.6 Kuratowski's Theorem.- 8.7 Hamiltonian Plane Graphs.- 8.8 Tait Coloring.- Notes.- IX Triangulated Graphs.- 9.0 Introduction.- 9.1 Perfect Graphs.- 9.2 Triangulated Graphs.- 9.3 Interval Graphs.- 9.4 Bipartite Graph B(G)ofa Graph G.- 9.5 Circular Arc Graphs.- 9.6 Exercises.- 9.7 Phasing of Traffic Lights at a Road Junction.- Notes.- X Applications.- 10.0 Introduction.- 10.1 The Connector Problem.- 10.2 Kruskal's Algorithm.- 10.3 Prim's Algorithm.- 10.4 Shortest-Path Problems.- 10.5 Timetable Problem.- 10.6 Application to Social Psychology.- 10.7 Exercises.- Notes.- List of Symbols.- References.
