Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T05:21:51.496Z Has data issue: false hasContentIssue false
  • English
  • Français

The Enumeration of Maps on the Torus and the Projective Plane

Published online by Cambridge University Press:  20 November 2018

E. A. Bender ,
E. R. Canfield  and
R. W. Robinson
E. A. Bender
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093
E. R. Canfield
Affiliation:
Department of Computer Science, University of Georgia Athens, Georgia 30602
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

Keywords

05C3005A15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 3 , 01 September 1988 , pp. 257 - 271
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Bender, E. A., Asymptotic methods in enumeration, SIAM Rev. 16 (1974), pp. 485–515 and 18 (1976), p. 292.Google Scholar
2. Bender, E. A. and Richmond, L. B., A survey of the asymptotic behaviour of maps, J. Combin. Theory Ser. B, to appear.Google Scholar
3. Bender, E. A. and Canfield, E. R., The asymptotic number of rooted maps on a surface, submitted.Google Scholar
4. Bender, E. A., Canfield, E. R. and Robinson, R. W., The asymptotic number of tree rooted maps on a surface, in preparation.Google Scholar
5. Brown, W. G., On the existence of square roots in certain rings of power series, Math. Ann. 170 (1976), pp. 327–333.Google Scholar
6. Brown, W. G., On the enumeration of non-planar maps, Mem. Amer. Math. Soc. 65 (1966).Google Scholar
7. Tutte, W. T., A census of planar maps, Canad. J. Math. 15 (1963) 249–271.Google Scholar
8. Tutte, W. T., On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 (1968), pp. 64–74.Google Scholar
9. Tutte, W. T., The enumerative theory of planar maps, A survey of Combinatorial Theory, (J. N. Srivastava et ai, éd.) North-Holland, Amsterdam (1972), pp. 437–448.Google Scholar
10. Walsh, T. R. S., and Lehman, A. B, Counting rooted maps by genus, I, J. Combin. Theory Ser. B 13(1972), pp. 192–218.Google Scholar
11. Walsh, T. R. S., and Lehman, A. B, Counting rooted maps by genus, II, J. Combin. Theory Ser. B 13 (1972), pp. 122–141. and 14 (1973), p. 185.Google Scholar
12. Walsh, T. R. S., and Lehman, A. B Counting rooted maps by genus, III: Nonseparable maps, J. Combin. Theory Ser. B 18(1975), pp. 222–259.Google Scholar
13. Wormald, N. C., private communication; see also [2, p. 25].Google Scholar