Independence Theory in Combinatorics An Introductory Account with Applications to Graphs and Transversals

Combinatorics may very loosely be described as that branch of mathematics which is concerned with the problems of arranging objects in accordance with various imposed constraints. It covers a wide range of ideas and because of its fundamental nature it has applications throughout mathematics. Among the well-established areas of combinatorics may now be included the studies of graphs and networks, block designs, games, transversals, and enumeration problem s concerning permutations and combinations, from which the subject earned its title, as weil as the theory of independence spaces (or matroids). Along this broad front,various central themes link together the very diverse ideas. The theme which we introduce in this book is that of the abstract concept of independence. Here the reason for the abstraction is to unify; and, as we sh all see, this unification pays off handsomely with applications and illuminating sidelights in a wide variety of combinatorial situations. The study of combinatorics in general, and independence theory in particular, accounts for a considerable amount of space in the mathematical journais. For the most part, however, the books on abstract independence so far written are at an advanced level, ·whereas the purpose of our short book is to provide an elementary in­ troduction to the subject.

1 Preliminaries.- 1.1 General introductory and historical remarks.- 1.2 Sets, families and graphs.- 1.3 Vector spaces; linear and affine independence.- Exercises.- 2 Independence spaces.- 2.1 Axioms and some basic theorems.- 2.2 Some induced structures.- 2.3 Submodular functions.- 2.4 Sums of independence structures.- Exercises.- 3 Graphic spaces.- 3.1 The cycle and cutset structures of a graph.- 3.2 Connections with vector spaces.- 3.3 Applications of independence theory to graphs.- Exercises.- 4 Transversal spaces.- 4.1 Hall’s theorem and its generalization.- 4.2 The partial transversals of a family of sets.- 4.3 Duals of transversal structures.- 4.4 Extensions of Hall’s theorem.- 4.5 Applications.- Exercises.- 5 Appendix on representability.- 5.1 Representability in general.- 5.2 Linear representability.- 5.3 Induced structures.- 5.4 Linear representability over specified fields.- 5.5 Some spaces which are not linearly representable.- Exercises.- Hints and solutions to the exercises.- Further reading.