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On Cycles and Connectivity in Planar Graphs

Published online by Cambridge University Press:  20 November 2018

M. D. Plummer  and
E. L. Wilson
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Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 283 - 288
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Barnette, D. and Jucovič, E., Hamiltonian circuits on 3-polytopes, J. Combinatorial Theory 9 (1970), 54–59.Google Scholar
2. Dirac, G., Généralisations du théorème de Menger, C. R. Acad. Sci. Paris, 250 (1960), 4252- 4253.Google Scholar
3. Dirac, G., In abstrakten Graphen vorhandene vollstandige 4-Graphen und ihre unterteilungen, Math. Nachr., 22 (1960), 61–85.Google Scholar
4. Harary, F., Graph theory, Addison-Wesley, Reading, Mass., 1969.Google Scholar
5. Ore, O., The four-color problem, Academic Press, New York, (1967), 68–74.Google Scholar
6. Tutte, W. T., A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956), 99–116.Google Scholar
7. Watkins, M. E., On the existence of certain disjoint arcs in graphs, Duke Math. J., 35 (1968), 231–246.Google Scholar
8. Watkins, M. E. and Mesner, D. M., Cycles and connectivity in graphs, Canad. J. Math., 19 (1967), 1319–1328.Google Scholar