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Almost all Convex Polyhedra are Asymmetric

Published online by Cambridge University Press:  20 November 2018

Edward A. Bender  and
Nicholas C. Wormald
Edward A. Bender
Affiliation:
University of California at San Diego, La Jolla, California
Nicholas C. Wormald
Affiliation:
University of Auckland, Auckland, New Zealand
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Although many types of rooted planar maps have been enumerated (see [8] for example), not much has been done on enumeration of entirely unrooted planar maps. Yet in virtually all cases of interest, it has appeared that comparatively very few of the maps are symmetric (have non-trivial automorphisms). This suggests that an asymptotic formula for the numbers of unrooted maps of a particular type on n edges can be obtained by dividing the numbers of rooted maps of that type on n edges by 4n, where 4n is the number of potentially distinct rootings of an asymmetric n-edged map. The assertion that almost all maps of a given type are asymmetric has previously been proved in only two non-trivial cases: for 3-connected planar triangulations by Tutte [9] and for all n-edged 3-connected planar maps in [5]. We prove here that it is also true for 3-connected planar maps with a given number of vertices and faces, uniformly as either parameter approaches infinity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 854 - 871
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Bender, E. A. and Richmond, L. B., The asymptotic enumeration of rooted convex polyhedra, J. Combinatorial Theory, Series B 36 (1984), 276–283.Google Scholar
2. Federico, P. J., The number of polyhedra, Philips Res. Rep. 30 (1975), 220–231.Google Scholar
3. Mani, P., Automorphismen von polyedrischen Graphen. Math. Ann. 192 (1971), 279–303.Google Scholar
4. Mullin, R. C. and Schellenberg, P. J., The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 3 (1968), 259–276.Google Scholar
5. Richmond, L. B. and Wormald, N. C., The asymptotic number of convex polyhedra, Trans. Amer. Math. Soc. 273 (1982), 721–735.Google Scholar
6. Tutte, W. T., A census of planar triangulations, Can. J. Math. 14 (1962), 21–38.Google Scholar
7. Tutte, W. T., A census of planar maps, Can. J. Math. 15 (1963), 249–271.Google Scholar
8. Tutte, W. T., The enumerative theory of planar maps, in A survey of combinatorial theory (North-Holland, Amsterdam, 1973), 437–448.Google Scholar
9. Tutte, W. T., On the enumeration of convex polyhdra, J. Combinatorial Theory, Series B, 28 (1980), 105–126.Google Scholar