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Some Asymptotical Estimates for Planar Eulerian Maps

Published online by Cambridge University Press:  12 September 2008

Valery A. Liskovets
Valery A. Liskovets
Affiliation:
Fakultäet Mathematik und Naturwiss, Technische Universitäet Dresden, Institut für Algebra, Mommsenstr. 13, D-01062 Dresden, Germany
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Abstract

In this paper, asymptotical estimates of the form Rn(1+o(1)) for various classes of planar valency-restricted Eulerian maps are established. It follows, in particular, that ‘almost all’ (as n → ∞) n-edged planar Eulerian maps have n/3 (1+o(1)) vertices. A brief survey of known asymptotical results (a table of values of R) for various classes of planar maps is also presented.

Information

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 2 , June 1996 , pp. 131 - 138
Copyright
Copyright © Cambridge University Press 1996

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References

[1]Bender, E. A. and Canfield, E. R. (1986) The asymptotic number of rooted maps on a surface, J. Combin. Th. A 43 244–257.Google Scholar
[2]Bender, E. A. and Canfield, E. R. (1993) The number of degree restricted rooted maps on the sphere, SIAM J. Discr. Math. 7 11–15.Google Scholar
[3]Bender, E. A. and Richmond, L. B. (1986) A survey of the asymptotic behaviour of maps, J. Combin. Th. B 40 297–329.CrossRefGoogle Scholar
[4]Brown, W. G. (1965) Enumeration of quadrangular dissections of the disk, Canad. J. Math. 17 302–317.CrossRefGoogle Scholar
[5]Goulden, I. P. and Jackson, D. P. (1983) Combinatorial Enumeration, John Wiley.Google Scholar
[6]Lando, S. K. and Zvonkin, A. K. (1992) Meanders, Selecta Math. Soviet. 11 117–144.Google Scholar
[7]Liskovets, V. A. (1985) Enumeration of nonisomorphic planar maps, Selecta Math. Soviet. 4 303–323.Google Scholar
[8]Liu, Y.-P. (1988) Enumeration of simple planar maps, Utilit. Math. 34 97–104.Google Scholar
[9]Liu, Y.-P.A note on the enumeration of bipartite planar maps, Acta Math. Scientia 12 185–188.Google Scholar
[10]Mullin, R. C. (1966) On the average number of trees in certain maps, Canad. J. Math. 18 33–41.CrossRefGoogle Scholar
[11]Tutte, W. T. (1962) A census of planar triangulations, Canad. J. Math. 14 21–38.Google Scholar
[12]Tutte, W. T. (1962) A census of slicings, Canad. J. Math. 14 708–722.CrossRefGoogle Scholar
[13]Tutte, W. T. (1963) A census of planar maps, Canad. J. Math. 15 249–271.CrossRefGoogle Scholar
[14]Tutte, W. T. (1984) Planar enumeration, Proc. Conf. ‘Graph Th. and Combin.’ (in honour of P. Erdös), Academic Press, London, pp. 315–319.Google Scholar
[15]Walsh, T. R. S. (1975) Hypermaps versus bipartite maps. J. Combin. Th. B 18 155–163.CrossRefGoogle Scholar
[16]Walsh, T. R. S. and Lehman, A. B. (1972) Counting rooted maps by genus I., J. Combin. Th. 13 192–218.CrossRefGoogle Scholar
[17]Walsh, T. R. S. and Lehman, A. B. (1975) Counting rooted maps by genus. III: Nonseparable maps, J. Combin. Th. 18 222–259.Google Scholar
[18]Yan, J.-Y. and Liu, Y.-P. (1991) Asymptotic enumerations for rooted planar maps and rooted outerplanar maps, Chinese Sci. Bull. 36 188–192.Google Scholar