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Cayley maps and symmetrical maps

Published online by Cambridge University Press:  24 October 2008

Norman Biggs
Norman Biggs
Affiliation:
Royal Holloway College, University of London
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Abstract

In this paper we shall show how combinatorial methods can be applied to the study of maps on orientable surfaces. Our main concern is with maps which possess a certain kind of symmetry, called vertex-transitivity. We show how an extension of the well-known method of Cayley can be used to construct such maps, and we give conditions which suffice for the automorphism groups of these maps to have non trivial vertex-stabilizers. Finally, we investigate the special case when the skeleton of the map is a complete graph; a classical theorem of Frobenius then implies that all vertex-transitive maps are given by our extension of Cayley's construction.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 381 - 386
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Biggs, N. L.Spanning trees of dual graphs. J. Combinatorial Theory, 11 (1971), 127–131.CrossRefGoogle Scholar
(2)Biggs, N. L.Automorphisms of imbedded graphs. J. Combinatorial Theory, 11 (1971), 132–138.CrossRefGoogle Scholar
(3)Burnside, W.Theory of groups of finite order (Cambridge, 1911).Google Scholar
(4)Gustin, W.Orientable imbeddings of Cayley graphs. Bull. Amer. Math. Soc. 69 (1963), 272–275.CrossRefGoogle Scholar
(5)Huppert, B.Endliche gruppen (Springer, 1967).CrossRefGoogle Scholar
(6)Wielandt, H.Finite permutation groups (Academic Press. N.Y. 1964).Google Scholar
(7)Youngs, J. W. T. The Heawood map colouring conjecture. In Graph theory and theoretical physics (Academic Press, London, 1967).Google Scholar