The number of polyhedral (-connected planar) graphs

Authors:
A. J. W. Duijvestijn and P. J. Federico

Journal:
Math. Comp. **37** (1981), 523-532

MSC:
Primary 05C30; Secondary 05C10, 52A25

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628713-3

MathSciNet review:
628713

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Abstract: Data is presented on the number of 3-connected 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 10-hedra to be presented for the first time, as well as estimates of the number of 11-hedra and dodecahedra.

**[1]**L. Euler, "Elementa doctrinae solidorum,"*Novi Comm. Acad. Petrop.*1752-3, v. 4, 1758, pp. 109-140;*Opera*(I), v. 26, pp. 71-93. Read November 25, 1750.**[2]**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.**[3]**T. P. Kirkman, "Application of the theory of the polyhedra to the enumeration and registration of resulte,"*Proc. Roy. Soc. London*, v. 12, 1862-3, pp. 341-380. Numbers: all classes with up to 8 faces or 8 vertices and class with 9 faces and 9 vertices.**[4]**O. Hermes, "Die Formen der Vielflache,"*J. Reine Angew. Math.*, [I], v. 120, 1899, pp. 27-59; [II], v. 120, 1899, pp. 305-353, plate 1; [III], v. 122, 1900, pp. 124-154, plates 1, 2; [IV], v. 123, 1901, pp. 312-342, 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].**[5]**M. Brückner,*Vielecke und Vielflache*, Teubner, Leipzig, 1900. Drawings: all trilinear (cubic) with up to 10 faces, folding plates 2-5. Figure 6 on plate 2 belongs with the 10-faced ones on plates 3-5.**[6]**C. J. Bouwkamp, "On the dissection of rectangles into squares. I,"*Nederl. Akad. Wetensch. Proc.*, v. A49, 1946, pp. 1776-1188 (=*Indag. Math.*, v. 8, 1946, pp. 724-736); II and III,*Nederl. Akad. Wetensch. Proc.*, v. A50, 1947, pp. 58-71, 72-78 (=*Indag. Math.*, v. 9, 1947, pp. 43-56, 57-63). Drawings: all with up to 14 edges, Part II, Figures 5, 7, 8. Professor Bouwkamp has called attention to the following corrections: Figure 5, second row, delete the second double arrow, third row, change to ; Figure 8, a missing dual pair can be reconstructed from the last row of data on page 71 [17]. MR**0019311 (8:398f)****[7]**C. J. Bouwkamp, A. J. W. Duijvestijn & P. Medema,*Table of c-Nets 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 3-connected planar graphs were called*c*-nete in the papers on squared rectangles, see [6], [9].**[8]**W. T. Tutte, "A theory of 3-connected graphs,"*Nederl. Akad. Wetensch. Proc. Ser.*A, v. 64 (=*Indag. Math.*, v. 23, 1961, pp. 451-455). MR**0140094 (25:3517)****[9]**A. J. W. Duijvestijn,*Electronic Computation of Squared Rectangles*, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962; also in*Philips Res. Rep.*, v. 17, 1962, pp. 523-612. Descriptions: 15 and 16 edges but only one of a dual pair. MR**0144492 (26:2036)****[10]**W. T. Tutte, "A census of planar maps,"*Canad. J. Math.*, v. 15, 1963, pp. 249-271. MR**0146823 (26:4343)****[11]**D. W. Grace,*Computer Search for Non-Isomorphic 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.**[12]**F. Harary & W. T. Tutte, "On the order of the group of a planar graph,"*J. Combin. Theory*, v. 1, 1966, pp. 394-395. MR**0200191 (34:90)****[13]**R. Bowen & S. Fiske, "Generation of triangulations of the sphere,"*Math. Comp.*, v. 21, 1967, pp. 250-252. Numbers: triangulations (duals of cubic) with up to 12 vertices. MR**0223277 (36:6325)****[14]**R. C. Mullin & P. J. Schellenberg, "The enumeration of*c*-nets via quadrangulations,"*J. Combin. Theory*, v. 4, 1968, pp. 259-276. In this paper, as well as in [10], "*c*-nets" refers to rooted 3-connected planar graphs, but in [7] the same term is used for the unrooted graphs. MR**0218275 (36:1362)****[15]**W. T. Tutte, "Counting planar maps,"*J. Recreational Math.*, v. 1, 1968, pp. 19-27.**[16]**P. J. Federico, "Enumeration of polyhedra: the number of 9-hedra,"*J. Combin. Theory*, v. 7, 1969, pp. 155-161. Numbers: to 9 faces or vertices and to 19 edges. MR**0243424 (39:4746)****[17]**C. J. Bouwkamp, Review of [7],*Math. Comp.*, v. 24, 1970, pp. 995-997.**[18]**F. Harary & E. M. Palmer,*Graphical Enumeration*, Academic Press, New York, 1973, p. 224. MR**0357214 (50:9682)****[19]**Doyle Britton & J. D. Dunitz, "A complete catalogue of polyhedra with eight or fewer vertices,"*Acta Cryst. Sect. A*, v. A29, 1973, pp. 362-371. Drawings: all classes with up to 8 vertices.**[20]**B. Grünbaum, "Polytopal graphs," in*Studies in Graph Theory*, Part II, MAA Studies in Mathematics, vol. 12, Math. Assoc. Amer., Washington, D. C., 1975, pp. 201-224. MR**0406868 (53:10654)****[21]**P. J. Federico, "The number of polyhedra,"*Philips Res. Rep.*, v. 30, 1975, pp. 220-231.**[22]**P. J. Federico, "Polyhedra with 4 to 8 faces,"*Geom. Dedicata*, v. 3, 1975, pp. 469-481. Drawings: all classes with up to 8 faces. The following corrections to the accompanying table are noted: line 24 change Symmetry entry to 4; line 169, change Faces entry to 242; line 219, change Faces entry to 224; line 301, change Vertices entry to 222. MR**0370358 (51:6585)****[23]**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.**[24]**A. J. W. Duijvestijn,*List of 3-Connected 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 self-dual 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.**[25]**W. T. Tutte, "On the enumeration of convex polyhedra,"*J. Combin. Theory Ser. B*, v. 28 B, 1980, pp. 105-126. MR**572468 (81j:05073)**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0628713-3

Article copyright:
© Copyright 1981
American Mathematical Society