Bipartite Graphs and their Applications ( Cambridge Tracts in Mathematics )

Publication series :Cambridge Tracts in Mathematics

Author: Armen S. Asratian; Tristan M. J. Denley; Roland Häggkvist  

Publisher: Cambridge University Press‎

Publication year: 1998

E-ISBN: 9780511899362

P-ISBN(Paperback): 9780521593458

Subject: O157.5 Graph

Keyword: 图解数学、图算数学

Language: ENG

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Bipartite Graphs and their Applications

Description

Bipartite graphs are perhaps the most basic of objects in graph theory, both from a theoretical and practical point of view. However, sometimes they have been considered only as a special class in some wider context. This book deals solely with bipartite graphs. Together with traditional material, the reader will also find many unusual results. Essentially all proofs are given in full; many of these have been streamlined specifically for this text. Numerous exercises of all standards have also been included. The theory is illustrated with many applications especially to problems in timetabling, chemistry, communication networks and computer science. For the most part the material is accessible to any reader with a graduate understanding of mathematics. However, the book contains advanced sections requiring much more specialized knowledge, which will be of interest to specialists in combinatorics and graph theory.

Chapter

2.3 Matrix characterisations of bipartite graphs Application

2.4 Gaussian elimination

Metric properties

3.1 Radius and diameter

3.2 Metric properties of trees

3.3 Metric properties of the n-cube Application

3.4 Addressing schemes for computer networks

Connectivity

4.1 k-connected graphs

4.2 k-edge-connected graphs Application

4.3 The construction of linear superconcentrators

Maximum matchings

5.1 Properties of maximum matchings

5.2 Finding a maximum matching

5.3 Maximum matchings in convex bipartite graphs

5.4 Stable matchings Application

5.5 The Generalised Assignment Problem

Expanding properties

6.1 Graphs with Hall's condition

6.2 Expanding graphs

6.3 Expanders Application

6.4 Expanders and sorters

Subgraphs with restricted degrees

7.1 (g,f)-factors

7.2 Subgraphs with given degrees

7.3 2-factors and Hamilton cycles

7.4 T-joins Application

7.5 Isomer problems in chemistry

Edge colourings

8.1 Edge colourings and timetables

8.2 Interval edge colourings

8.3 List colourings

8.4 Colour-feasible sequences

8.5 Transformations of proper colourings

8.6 Uniquely colourable bipartite graphs Application

8.7 Rearrangeable telephone networks

Doubly stochastic matrices and bipartite graphs

9.1 Convex representations of doubly stochastic matrices

9.2 Matrices with a unique convex representation

9.3 Permanents and perfect matchings

Coverings

10.1 Some examples of covering problems

10.2 Vertex coverings and independent sets

10.3 Dulmage and Mendelsohn's canonical decomposition Application

10.4 Decomposition of partially ordered sets into chains

Some combinatorial applications

11.1 Systems of distinct representatives

11.2 Generation of subsets of a set

11.3 Pebbling in hypergrids

11.4 Completing latin squares

Bipartite subgraphs of arbitrary graphs

12.1 Spanning bipartite subgraphs

12.2 Covering the edges of a graph with bipartite subgraphs Applications

12.3 Optimal spanning trees and the Travelling Salesman Problem

12.4 The optimal spanning tree and optimal path problems

Appendix

References

Index

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