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On the Enumeration of Tree-Rooted Maps

Published online by Cambridge University Press:  20 November 2018

R. C. Mullin
R. C. Mullin*
Affiliation:
University of Waterloo
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It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 174 - 183
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Harary, F. and Prins, G., The number of homeomorphically irreducible trees and other species, Acta. Math., 101 (1959), 141–162.Google Scholar
2. Mullin, R. C., On the average number of Hamiltonian polygons in a rooted triangular map, Pacific J. Math., 16 (1966), 139–145.Google Scholar
3. Mullin, R. C., On the average number of trees in certain maps, Can. J. Math., 18 (1966), 33–41.Google Scholar
4. Tutte, W. T., A census of planar maps, Can. J. Math., 15 (1963), 249–271.Google Scholar
5. Tutte, W. T., The number of planted plane trees with a given partition, Amer. Math. Monthly, 71 (1964), 272–277.Google Scholar
6. Williamson, B., Differential calculus (London, 1889).Google Scholar