The number of polyhedral ($3$connected planar) graphs
Authors:
A. J. W. Duijvestijn and P. J. Federico
Journal:
Math. Comp. 37 (1981), 523532
MSC:
Primary 05C30; Secondary 05C10, 52A25
DOI:
https://doi.org/10.1090/S00255718198106287133
MathSciNet review:
628713
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: Data is presented on the number of 3connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 22 edges. The numbers of such graphs having the same number of edges, and the same number of vertices and faces, are tabulated. Conjectured asymptotic formulas by W. T. Tutte and by R. C. Mullin and P. J. Schellenberg are discussed. Additional data beyond 22 edges are given enabling the number of 10hedra to be presented for the first time, as well as estimates of the number of 11hedra and dodecahedra.

L. Euler, "Elementa doctrinae solidorum," Novi Comm. Acad. Petrop. 17523, v. 4, 1758, pp. 109140; Opera (I), v. 26, pp. 7193. Read November 25, 1750.
J. Steiner, "Problème de situation," Ann. de Math., v. 19, 1828, p. 36; Gesammelte Werke, vol. 1, p. 227. Descriptions: 4, 5 and 6 faces.
T. P. Kirkman, "Application of the theory of the polyhedra to the enumeration and registration of resulte," Proc. Roy. Soc. London, v. 12, 18623, pp. 341380. Numbers: all classes with up to 8 faces or 8 vertices and class with 9 faces and 9 vertices.
O. Hermes, "Die Formen der Vielflache," J. Reine Angew. Math., [I], v. 120, 1899, pp. 2759; [II], v. 120, 1899, pp. 305353, plate 1; [III], v. 122, 1900, pp. 124154, plates 1, 2; [IV], v. 123, 1901, pp. 312342, plate 1. Descriptions: all with up to 8 faces, Part II; all trilinear (cubic) with up to 10 faces, Part I. Contains erroneous tables for 9 faces and 9 vertices and erroneous numbers for trilinear (cubic) with 11 and 12 faces [16].
M. Brückner, Vielecke und Vielflache, Teubner, Leipzig, 1900. Drawings: all trilinear (cubic) with up to 10 faces, folding plates 25. Figure 6 on plate 2 belongs with the 10faced ones on plates 35.
 C. J. Bouwkamp, On the dissection of rectangles into squares. II, III, Nederl. Akad. Wetensch., Proc. 50 (1947), 58–71, 72–78 = Indagationes Math. 9, 43–56, 57–63 (1947). MR 19311 C. J. Bouwkamp, A. J. W. Duijvestijn & P. Medema, Table of cNets of Orders 8 to 19, Inclusive, Philips Research Laboratories, Eindhoven, Netherlands, 2 vols., 1960. Unpublished available in UMT file. Descriptions: all with 8 to 19 edges except that only one of a dual pair is listed. See [17] for description. The 3connected planar graphs were called cnete in the papers on squared rectangles, see [6], [9].
 W. T. Tutte, A theory of $3$connected graphs, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 441–455. MR 0140094
 Adrianus Johannes Wilhelmus Duijvestijn, Electronic computation of squared rectangles, Thesis, Technische Wetenschap aan de Technische Hogeschool te Eindhoven, Eindhoven, 1962. MR 0144492
 W. T. Tutte, A census of planar maps, Canadian J. Math. 15 (1963), 249–271. MR 146823, DOI https://doi.org/10.4153/CJM1963029x D. W. Grace, Computer Search for NonIsomorphic Convex Polyhedra, Report CS15, Computer Sci. Dept., Stanford Univ., 1965 (copy obtainable from National Technical Information Service, Dept. of Commerce, Springfield, Va. 22151 as Document AD611, 366). Descriptions: trilinear (cubic) with up to 11 faces.
 Frank Harary and William T. Tutte, On the order of the group of a planar map, J. Combinatorial Theory 1 (1966), 394–395. MR 200191
 Rufus Bowen and Stephen Fisk, Generations of triangulations of the sphere, Math. Comp. 21 (1967), 250–252. MR 223277, DOI https://doi.org/10.1090/S00255718196702232773
 R. C. Mullin and P. J. Schellenberg, The enumeration of $c$nets via quadrangulations, J. Combinatorial Theory 4 (1968), 259–276. MR 218275 W. T. Tutte, "Counting planar maps," J. Recreational Math., v. 1, 1968, pp. 1927.
 P. J. Federico, Enumeration of polyhedra: The number of $9$hedra, J. Combinatorial Theory 7 (1969), 155–161. MR 243424 C. J. Bouwkamp, Review of [7], Math. Comp., v. 24, 1970, pp. 995997.
 Frank Harary and Edgar M. Palmer, Graphical enumeration, Academic Press, New YorkLondon, 1973. MR 0357214 Doyle Britton & J. D. Dunitz, "A complete catalogue of polyhedra with eight or fewer vertices," Acta Cryst. Sect. A, v. A29, 1973, pp. 362371. Drawings: all classes with up to 8 vertices.
 Branko Grünbaum, Polytopal graphs, Studies in graph theory, Part II, Math. Assoc. Amer., Washington, D. C., 1975, pp. 201–224. Studies in Math., Vol. 12. MR 0406868 P. J. Federico, "The number of polyhedra," Philips Res. Rep., v. 30, 1975, pp. 220$^{\ast }$231$^{\ast }$.
 P. J. Federico, Polyhedra with $4$ to $8$ faces, Geometriae Dedicata 3 (1974/75), 469–481. MR 370358, DOI https://doi.org/10.1007/BF00181378 A. J. W. Duijvestijn, Algorithmic Calculation of the Order of the Automorphism Group of a Graph, Memorandum No. 221, Twente Univ. of Technology, Enschede, Netherlands, 1978. A. J. W. Duijvestijn, List of 3Connected Planar Graphs with 6 to 22 Edges, Twente Univ. of Technology, Enschede, Netherlands, 1979. (Computer tape.) Descriptions: These are arranged in files, each file containing graphs of the same number of edges ordered by identification number. Only one of a dual pair is listed the one with fewer vertices than faces; if the number of these is the same for a dual pair, the one with the larger identification number is listed. The graphs are coded by lettering the vertices A, B, C, D, ... and giving the circuit of vertices for each face, with a separation mark. Each entry gives first the code of the graph and then follows in order, an indication whether the graph is selfdual or not, the order of the automorphism group, and the identification number. Arrangements for obtaining a copy of the tape can be made by communicating with the author.
 W. T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105–126. MR 572468, DOI https://doi.org/10.1016/00958956%2880%29900593
Retrieve articles in Mathematics of Computation with MSC: 05C30, 05C10, 52A25
Retrieve articles in all journals with MSC: 05C30, 05C10, 52A25
Additional Information
Article copyright:
© Copyright 1981
American Mathematical Society