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Heawood's theorem and connectivity

Part of: Graph theory

Published online by Cambridge University Press:  26 February 2010

R. J. Cook
R. J. Cook
Affiliation:
University College, Cardiff
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Extract

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K(G) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ(G) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph Kp we have

MSC classification

Secondary: 05C99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 201 - 207
Copyright
Copyright © University College London 1973

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References

1. Beineke, L. W. and Harary, F., “Inequalities involving the genus of a graph and its thickness”, Proc. Glasgow Math. Assoc., 7 (1965), 19–21.CrossRefGoogle Scholar
2. Chartrand, G. and Harary, F., “Graphs with prescribed connectivities”, Theory of graphs (Erdös, P. and Katona, G., eds.), Akademiai Kiado (budapest, 1968), 61–63.Google Scholar
3. Cook, R. J., “Chromatic number and girth,” to appear in Periodica Math. Hungarica.Google Scholar
4. Harary, F., Graph theory (Addison-Wesley, Reading, Mass., 1969).CrossRefGoogle Scholar
5. Heawood, P. J., “Map colour theorem”, Quart. J. Math., 24 (1890), 332–338.Google Scholar
6. Kelly, J. B. and Kelly, L. M., “Paths and circuits in critical graphs”, Amer. J. Math., 76 (1954), 786–792.CrossRefGoogle Scholar
7. Menger, K., “Zur allgemeinen Kurventheorie”, Fund. Math., 10 (1927), 96–115.CrossRefGoogle Scholar
8. Ringel, G., Färbungsprobleme auf Flachen und Graphen, Deutscher Verlag der Wissenschaften, Berlin, 1962.Google Scholar
9. Ringel, G., and Youngs, J. W. T., “Solution of the Heawood map-coloring problem”, Proc. Nat. Acad. Sci. USA, 60 (1968), 438–445.CrossRefGoogle ScholarPubMed
10. Turan, P.,“Eine Extremalaufgabe aus der Graphentheorie”, Mat. Fiz. Lapok, 48 (1941), 436–452.Google Scholar
11. Whitney, H., “Congruent graphs and the connectivity of graphs”, Amer. J. Math., 54 (1932), 150–168.CrossRefGoogle Scholar
12. Zykov, A. A., “On some properties of linear complexes” (Russian), Mat. Zbornik, 24 (1949), 163–188.Google Scholar