Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-15T10:17:04.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Enumerating Dissectible Polyhedra by Their Automorphism Groups

Published online by Cambridge University Press:  20 November 2018

L. W. Beineke  and
R. E. Pippert
L. W. Beineke
Affiliation:
Purdue University at Fort Wayne, Fort Wayne, Indiana
R. E. Pippert
Affiliation:
Western Michigan University, Kalamazoo, Michigan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 1 , February 1974 , pp. 50 - 67
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Beineke, L. W. and Pippert, R. E., The number of labeled dissections of a k-ball, Math. Ann. 191 (1971), 87–98.Google Scholar
2. Beineke, L. W. and Pippert, R. E., A census of ball and disk dissections, Chapter in Graph Theory and Applications (Proceedings of the Conference on Graph Theory and Applications, May, 1972), Y. Alavi, D. R. Lick, and A. T. White, Eds. Springer-Verlag, New York, 1972, pp. 25–40.Google Scholar
3. Brown, W. G., Enumeration of non-separable planar maps, Can. J. Math. 15 (1963), 526–545.Google Scholar
4. Brown, W. G., Historial note on a recurrent combinatorial problem, Amer. Math. Monthly 72 (1965), 973–977.Google Scholar
5. Brown, W. G. and Tutte, W. T., On the enumeration of rooted non-separable planar maps, Can. J. Math. 16 (1964), 572–577.Google Scholar
6. Coxeter, H. S. M., Introduction to geometry (Wiley, New York, 1961).Google Scholar
7. Euler, L., Novi commentarii academiae scientiarum imperialis petropolitanae 7 (1758-1759), 13-14.Google Scholar
8. Guy, R. K., Dissecting a polygon into triangles, Bull. Malayan Math. Soc. 5 (1958), 57–60. Same title, Research Paper No. 9, The University of Calgary, 1967.Google Scholar
9. Moon, J. W. and Moser, L., Triangular dissections of n-gons, Can. Math. Bull. 6 (1963), 175–178.Google Scholar
10. Takeo, F., On triangulated graphs. I, Bull. Fukuoka Univ. Ed. III 10 (1960), 9–21.Google Scholar
11. Takeo, F., On triangulated graphs. II, Bull. Fukuoka Univ. Ed. III 11 (1961), 17–31.Google Scholar
12. Takeo, F., On triangulated graphs. III, Bull. Fukuoka Univ. Ed. III 13 (1963), 11–21.Google Scholar
13. Takeo, F., On triangulated graphs. IV, Bull. Fukuoka Univ. Ed. III 14 (1964), 19–30.Google Scholar
14. Tutte, W. T., A census of planar triangulations, Can. J. Math. 14 (1962), 21–38.Google Scholar
15. Tutte, W. T., A census of planar maps, Can. J. Math. 15 (1963), 249–271.Google Scholar