Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T00:37:36.177Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Average Number of Trees in Certain Maps

Published online by Cambridge University Press:  20 November 2018

R. C. Mullin
R. C. Mullin*
Affiliation:
University of Waterloo
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.

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.

Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 33 - 41
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Brown, W. G., Enumeration of triangulations of the disc, Proc. London Math. Soc, 14 (1964), 746–768.Google Scholar
2. Etherington, M. H., Some problems in non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1–6.Google Scholar
3. Harary, F., Prins, G., and Tutte, W. T., The number of plane trees, Koninkl. Nederl. Akad. Wetensch. Proc, A, 67 and Indag Math., 26 (1964), 319–329.Google Scholar
4. Mullin, R. C., Enumeration of rooted triangular maps, Amer. Math. Monthly, 71 (1964), 1007–1010.Google Scholar
5. Mullin, R. C., On counting rooted triangular maps, Can. J. Math., 17 (1965), 373–382.Google Scholar
6. Mullin, R. C., On the average number of Hamiltonian polygons in a rooted triangular map, Pacific J. Math, (to appear).Google Scholar
7. Tutte, W. T., A census of planar maps, Can. J. Math., 15 (1963), 249–271.Google Scholar
8. Whittaker, E. T. and Watson, G. N., A Course of modern analysis (Cambridge, 1940).Google Scholar