Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-01T04:10:47.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

Published online by Cambridge University Press:  20 November 2018

J. A. Bondy  and
R. C. Entringer
J. A. Bondy
Affiliation:
University of Waterloo, Waterloo, Ontario
R. C. Entringer
Affiliation:
University of New Mexico, Albuquerque, New Mexico
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].

Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1325 - 1332
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bermond, J. C., On Hamiltonian walks, Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975). Congressus Numerantium, Utilitas Math. Winnipeg, Man. 15 (1976), 41–51.Google Scholar
2. Bondy, J. A., Large cycles in graphs, Discrete Math. 1.2 (1971), 121–132.Google Scholar
3. Bondy, J. A., Hamilton cycles in graphs and digraphs, Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Florida (1978), Utilitas Math. 21 (1978), 3–28.Google Scholar
4. Bondy, J. A. and Simonovits, M., Longest cycles in 3-connected S-regular graphs, Can. J. Math., to appear.CrossRefGoogle Scholar
5. Dirac, G. A., Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69–81.Google Scholar
6. Dirac, G. A., Palhs and circuits in graphs: extreme cases, Acta Math. Acad. Sci. Hungar. 10 (1959), 357–362.Google Scholar
7. Erdös, P. and Gallai, T., On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.Google Scholar
8. Grötschel, M., Graphs with cycles containing given paths, Ann. of Discrete Math. 1 (1977), 233–245.Google Scholar
9. Jackson, B., Hamilton cycles in regular 2-connected graphs, J. Combinatorial Theory Ser. B. 29 (1980), 47–67.Google Scholar
10. Lang, R. and Walther, H., Über längste Kreise in regulären Graphen, Beiträge zur Graphentheorie (Kolloquium, Manebach, 1967). Teubner, Leipzig (1968), 91–98.Google Scholar
11. Linial, N., A lower bound for the circumference of a graph, Discrete Math. 15.3 (1976), 297–300.Google Scholar
12. Ore, O., On a graph theorem by Dirac, J. Combinatorial Theory 2 (1967), 383–392.Google Scholar
13. Pósa, L., On the circuits of finite graphs, (Russian summary) Magyar Tud. Akad. Mat. Kutato Int. Közl. 8 (1963), 355–361.Google Scholar
14. Voss, H.-J., Maximal circuits and paths in graphs: extreme cases, Combinatorics (Proc. Conf. Keszthely, 1976), Colloq. Math. Soc. János Bolyai (North-Holland Publishing Company, New York) 18 (1978), 1099–1122.Google Scholar
15. Voss, H.-J., Bridges of longest circuits and of longest paths in graphs, Beiträge zur Graphentheorie und deren Anwendungen (Intern. Kolloquium, Oberhof, 1977), 275–286.Google Scholar
16. Voss, H.-J. and Zuluaga, C., Maximale gerade und ungerade Kreise in Graphen, I, Wiss. Z. Techn. Hochsch. Ilmenau 23A (1977), 57–70.Google Scholar