Combinatorics with Emphasis on the Theory of Graphs

Description

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 Front Matter....Pages i-xv Finite Sets....Pages 1-27 Algebraic Structures on Finite Sets....Pages 28-56 Multigraphs....Pages 57-97 Networks....Pages 98-125 Matchings and Related Structures....Pages 126-152 Separation and Connectivity in Multigraphs....Pages 153-177 Chromatic Theory of Graphs....Pages 178-212 Two Famous Problems....Pages 213-229 Designs....Pages 230-264 Matroid Theory....Pages 265-309 Enumeration Theory....Pages 310-335 Back Matter....Pages 337-351