Almost all Convex Polyhedra are Asymmetric

Abstract

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.