Combinatorics with Emphasis on the Theory of Graphs: (54 Graduate Texts in Mathematics)

Combinatorics and graph theory have mushroomed in recent years. Many overlapping or equivalent results have been produced. Some of these are special cases of unformulated or unrecognized general theorems. The body of knowledge has now reached a stage where approaches toward unification are overdue. To paraphrase Professor Gian-Carlo Rota (Toronto, 1967), "Combinatorics needs fewer theorems and more theory. " In this book we are doing two things at the same time: A. We are presenting a unified treatment of much of combinatorics and graph theory. We have constructed a concise algebraically­ based, but otherwise self-contained theory, which at one time embraces the basic theorems that one normally wishes to prove while giving a common terminology and framework for the develop­ ment of further more specialized results. B. We are writing a textbook whereby a student of mathematics or a mathematician with another specialty can learn combinatorics and graph theory. We want this learning to be done in a much more unified way than has generally been possible from the existing literature. Our most difficult problem in the course of writing this book has been to keep A and B in balance. On the one hand, this book would be useless as a textbook if certain intuitively appealing, classical combinatorial results were either overlooked or were treated only at a level of abstraction rendering them beyond all recognition.

Table of Contents:
I Finite Sets.- IA Conventions and Basic Notation.- IB Selections and Partitions.- IC Fundamentals of Enumeration.- ID Systems.- IE Parameters of Systems.- II Algebraic Structures on Finite Sets.- IIA Vector Spaces of Finite Sets.- IIB Ordering.- IIC Connectedness and Components.- IID The Spaces of a System.- IIE The Automorphism Groups of Systems.- III Multigraphs.- IIIA The Spaces of a Multigraph.- IIIB Biconnectedness.- IIIC Forests.- IIID Graphic Spaces.- IIIE Planar Multigraphs.- IIIF Euler’s Formula.- IIIG Kuratowski’s Theorem.- IV Networks.- IVA Algebraic Preliminaries.- IVB The Flow Space.- IVC Max-Flow—Min-Cut.- IVD The Flow Algorithm.- IVE The Classical Form of Max-Flow—Min-Cut.- IVF The Vertex Form of Max-Flow—Min-Cut.- IVG Doubly-Capacitated Networks and Dilworth’s Theorem.- V Matchings and Related Structures.- VA Matchings in Bipartite Graphs.- VB 1-Factors.- VC Coverings and Independent Sets in Graphs.- VD Systems with Representatives.- VE 0, 1-Matrices.- VF Enumerative Considerations.- VI Separation and Connectivity in Multigraphs.- VIA The Menger Theorem.- VIB Generalizations of the Menger Theorem.- VIC Connectivity.- VID Fragments.- VIE Tutte Connectivity and Connectivity of Subspaces.- VII Chromatic Theory of Graphs.- VIIA Basic Concepts and Critical Graphs.- VIIB Chromatic Theory of Planar Graphs.- VIIC The Imbedding Index.- VIID The Euler Characteristic and Genus of a Graph.- VIIE The Edmonds Imbedding Technique.- VIII Two Famous Problems.- VIIIA Cliques and Scatterings.- VIIIB Ramsey’s Theorem.- VIIIC The Ramsey Theorem for Graphs.- VIIID Perfect Graphs.- IX Designs.- IXA Parameters of Designs.- IXB Design-Types.- IXC t-Designs.- IXD Finite Projective Planes.- IXE Partially-Balanced Incomplete Block Designs.- IXF Partial Geometries.- XMatroid Theory.- XA Exchange Systems.- XB Matroids.- XC Rank and Closure.- XD Orthogonality and Minors.- XE Transversal Matroids.- XF Representability.- XI Enumeration Theory.- XIA Formal Power Series.- XIB Generating Functions.- XIC Pólya Theory.- XID Möbius Functions.- Index of Symbols.