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Bi-embeddings of graphs

Published online by Cambridge University Press:  18 May 2009

I. Anderson  and
R. J. Cook
I. Anderson
Affiliation:
University of Glasgow, Glasgow G12 8QW
R. J. Cook
Affiliation:
University College Cardiff
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Let γ and γ' be non-negative integers. We say that the graph G is (γ, γ') bi-embeddable if G can be embedded in a surface of genus γ and the complement Ḡ of G can be embedded in a surface of genus γ'. Let N(γ, γ') be the least integer such that every graph with at least N(γ, γ') points is not (γ, γ') bi-embeddable. It has been shown in [1] and [5] that N(0, 0) = 9; this result was also obtained by John R. Ball of the Carnegie Institute of Technology. Our object here is to obtain upper and lower bounds for N(γ, γ').

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 2 , September 1974 , pp. 162 - 165
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Battle, J., Harary, F. and Kodama, Y., Every planar graph with nine points has a non-planar complement, Bull. Amer. Math. Soc. 68 (1962), 569–571.CrossRefGoogle Scholar
2.Battle, J., Harary, F., Kodama, Y. and Youngs, J. W. T., Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), 565–568.CrossRefGoogle Scholar
3.Ringel, G., Das Geschlecht des vollständigen paaren Graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139–150.CrossRefGoogle Scholar
4.Ringel, G. and Youngs, J. W. T., Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 438–445.CrossRefGoogle ScholarPubMed
5.Tutte, W. T., On the non-biplanar character of the complete 9-graph, Canad. Math. Bull. 6 (1963), 319–330.CrossRefGoogle Scholar
6.White, A. T., The genus of the complete tripartite graph Kmn,n,n, J. Combinatorial Theory 7 (1969), 283–285.CrossRefGoogle Scholar
7.Youngs, J. W. T., Minimal imbeddings and the genus of a graph, J. Math. Mech. 12 (1963), 303–315.Google Scholar