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Combinatorial Oriented Maps

Published online by Cambridge University Press:  20 November 2018

W. T. Tutte
W. T. Tutte*
Affiliation:
University of Waterloo, Waterloo, Ontario
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An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.

On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S. The choice of P rather than P –l, which corresponds to the other sense of rotation, makes the map “oriented”.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979, pp. 986 - 1004
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Brahana, H. R., Systems oJ circuits on two-dimensional manifolds, Ann. Math., (2). 23 (1921), 144–168.Google Scholar
2. Cori, R., Graphes planaires et systèmes de parenthèses, Centre National de la Recherche Scientifique, Institut Biaise Pascal, (1969).Google Scholar
3. Cori, R., Un code pour les graphes pla —aires et ses applications, Asterisk 27 (1975).Google Scholar
4. Edmonds, J. R., A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc, 7 (1960), 646.Google Scholar
5. Jacques, A., Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Pari. 267 (1968), 625–627.Google Scholar
6. Jacques, A., Constellations et graphes topologiques, In Combinatorial Theory and its Applications II, Budapest (1970).Google Scholar
7. Tutte, W. T., Connectivity in graphs (University of Toronto Press, 1966).Google Scholar
8. Walsh, T. R. S., Combinatorial enumeration of non-planar maps, Thesis, U. of Toronto (1971).Google Scholar
9. Walsh, T. R. S. and Lehman, A. B., Counting rooted maps by genus I, J. Combinatorial Theor. 13 (1972), 192–218.Google Scholar