One-Factorizations

This book has grown out of graduate courses given by the author at Southern Illinois University, Carbondale, as well as a series of seminars delivered at Curtin University of Technology, Western Australia. The book is intended to be used both as a textbook at the graduate level and also as a professional reference. The topic of one-factorizations fits into the theory of combinatorial designs just as much as it does into graph theory. Factors and factorizations occur as building blocks in the theory of designs in a number of places. Our approach owes as much to design theory as it does to graph theory. It is expected that nearly all readers will have some background in the theory of graphs, such as an advanced undergraduate course in Graph Theory or Applied Graph Theory. However, the book is self-contained, and the first two chapters are a thumbnail sketch of basic graph theory. Many readers will merely skim these chapters, observing our notational conventions along the way. (These introductory chapters could, in fact, enable some instructors to Ilse the book for a somewhat eccentric introduction to graph theory.) Chapter 3 introduces one-factors and one-factorizations. The next two chapters outline two major application areas: combinatorial arrays and tournaments. These two related areas have provided the impetus for a good deal of study of one-factorizations.

Inhalt
1 Graphs.- 2 Walks, Paths and Cycles.- 3 One-Factors and One-Factorizations.- 4 Orthogonal One-Factorizations.- 5 Tournament Applications of One-Factorizations.- 6 A General Existence Theorem.- 7 Graphs without One-Factors.- 8 Edge-Colorings.- 9 One-Factorizations and Triple Systems.- 10 Starters.- 11 Invariants of One-Factorizations.- 12 Automorphisms and Asymptotic Numbers of One-Factorizations.- 13 Systems of Distinct Representatives.- 14 Subfactorizations and Asymptotic Numbers of One-Factorizations.- 15 Cyclic One-Factorizations.- 16 Perfect Factorizations.- 17 One-Factorizations of Multigraphs.- 18 Maximal Sets of Factors.- 19 The One-Factorization Conjecture.- 20 Premature Sets of Factors.- 21 Cartesian Products.- 22 Kotzig's Problem.- 23 Other Products.- B Generators of Simple Indecomposable Factorizations.- C Generators of Nonsimple Indecomposable Factorizations.- References.