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For more than one hundred years, the development of graph theory was inspired and guided mainly by the Four-Colour Conjecture. The resolution of the conjecture by K. Appel and W. Haken in 1976, the year in which our first book Graph Theory with Applications appeared, marked a turning point in its history. Since then, the subject has experienced explosive growth, due in large measure to its role as an essential structure underpinning modern applied mathematics. Computer science and combinatorial optimization, in particular, draw upon and contribute to the development of the theory of graphs. Moreover, in a world where communication is of prime importance, the versatility of graphs makes them indispensable tools in the design and analysis of communication networks.Building on the foundations laid by Claude Berge, Paul Erd˝os, Bill Tutte, and others, a new generation of graph-theorists has enriched and transformed the subject by developing powerful new techniques, many borrowed from other areas of mathematics. These have led, in particular, to the resolution of several longstanding conjectures, including Berge’s Strong Perfect Graph Conjecture and Kneser’s Conjecture, both on colourings, and Gallai’s Conjecture on cycle coverings.
Content:-
1. Graphs
2. Subgraphs
3. Connected Graphs
4. Trees
5. Nonseparable Graphs
6. Tree-Search Algorithms
7. Flows in Networks
8. Complexity of Algorithms
9. Connectivity
10. Planar Graphs
11. The Four-Colour Problem
12. Stable Sets and Cliques
13. The Probabilistic Method
14. Vertex Colourings
15. Colourings of Maps
16. Matchings
17. Edge Colourings
18. Hamilton Cycles
19. Coverings and Packings in Directed Graphs
20. Electrical Networks
21. Integer Flows and Coverings
Unsolved Problems
References
General Mathematical Notation
Graph Parameters
Operations and Relations
Families of Graphs
Structures
Other Notation
Index
Author Details
"J.A. Bondy"
"U.S.R. Murty"
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