Abstract
Let the vertices of a 2n-gon be labelled, in an order of traversal: 0, 1′, 1, 2′, 2, ..., (n−1)′, n−1, 0′. Let G(n,m) denote the bicubic graph derived from this 2n-gon by adjunction of the chords (i,(i+m)′), i = 0, 1, 2, ..., n−1, the addition being taken modulo n. Restricting ourselves to the case when n is prime, we determine the isomorphism classes of the graphs G(n,m), and the corresponding automorphism groups. Various applications are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bouwer, I. Z., and Frucht, R., "Line Minimal Graphs with Cyclic Groups," A Survey of Combinatorial Theory, (Ed. J. N. Srivastava), North-Holland, New York, 1973, 53–67.
Coxeter, H. S. M., "Self-dual Configurations and Regular Graphs," Bull. Amer. Math. Soc. 56 (1950), 413–455.
Coxeter, H. S. M. and Moser, W. O. J., "Generators and Relations for Discrete Groups," 3rd Edition, Springer-Verlag, New York, 1972, Section 8.4, 107–109.
Foster, R. M., "A Census of Trivalent Symmetrical Graphs I," presented at the conference on Graph Theory and Combin. Analysis, Waterloo, Ontario, 1966.
Frucht, R., "A One-regular Graph of Degree Three," Canad. J. Math. 4 (1952), 240–247.
Frucht, R., "How to Describe a Graph," Ann. N. Y. Ac. Sci. 175, Part I (1970), 159–167.
Harary, F., "Graph Theory," Addison-Wesley, Reading, Mass., 1969.
Miller, R. C., "The Trivalent Symmetric Graphs of Girth at Most 6", J. Combinatorial Theory, Series B, 10 (1971), 163–182.
Tutte, W. T., "A Family of Cubical Graphs," Proc. Cambridge Philos. Soc. 43 (1947), 459–474.
Watkins, M. E., "On the Action of Non-abelian Groups on Graphs," J. Combinatorial Theory 11 (1971), 95–104.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin
About this paper
Cite this paper
Boreham, T.G., Bouwer, I.Z., Frucht, R.W. (1974). A useful family of bicubic graphs. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066443
Download citation
DOI: https://doi.org/10.1007/BFb0066443
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06854-9
Online ISBN: 978-3-540-37809-9
eBook Packages: Springer Book Archive