Fractional Graph Theory

A Rational Approach to the Theory of Graphs

Edward R. Scheinerman, Daniel H. Ullman

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486292137
• Topic: Mathematics
• Subtopic: Discrete Mathematics
• Index: Mathematics

A unified treatment of the most important results in the study of fractional graph concepts, this volume explores the various ways in which integer-valued concepts can be modified to derive nonintegral values. It begins with the general fractional theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics. Subjects include fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, and fractional isomorphism. The final chapter examines additional topics such as fractional domination, fractional intersection numbers, and fractional aspects of partially ordered sets.
Challenging exercises reinforce the contents of each chapter, and the authors provide substantial references and bibliographic materials. A comprehensive reference for researchers, this volume also constitutes an excellent graduate-level text for students of graph theory and linear programming.

1

General Theory: Hypergraphs

1.1 Hypergraph covering and packing

1.2 Fractional covering and packing

1.3 Some consequences

1.4 A game-theoretic approach

1.5 Duality and duality

1.6 Asymptotic covering and packing

1.7 Exercises

1.8 Notes

Our purpose is to reveal the rational side of graph theory: We seek to convert integer-based definitions and invariants into their fractional analogues. When we do this, we want to be sure that we have formulated the “right” definitions—conversions from integer to real that are, in some sense, natural. Here are two ways we might judge if an integer-to-real conversion process is natural: First, when two seemingly disparate conversions yield the same concept, then we feel confident that the fractionalized concept is important (and not tied to how we made the conversion). Second, when the same conversion process works for a variety of concepts (e.g., we convert from matching number and chromatic number to their fractional analogues by the same methods), then we feel we have arrived at a reasonable way to do the integer-to-real transformation. Happily, we can often satisfy both of these tests for “naturalness”. If a variety of attractive theorems arise from the new definition, then we can be certain that we ...