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Counting rooted non-separable nearly cubic planar maps. (English) Zbl 1513.05202

In such papers as [W. T. Tutte, Can. J. Math. 14, 21–38 (1962; Zbl 0103.39603); W. G. Brown and W. T. Tutte, Can. J. Math. 16, 572–577 (1964; Zbl 0119.38804)] W. T. Tutte and his collaborators and students developed procedures for the counting of various classes of “rooted” planar maps according to various parameters. Those procedures often involved finding solutions in terms of convenient parameters, following which explicit solutions might be found by Lagrangian inversion, where the numbers determined often involved multiple summations; the rooting usually involved the fixing of half of an edge, as well as its relation to a specific face of the map with which it is incident. The author was a coauthor of the paper [J. Cai and Y. Liu, Discrete Math. 207, No. 1–3, 9–24 (1999; Zbl 0934.05074)], and, in the present paper, he claims that some results and methods used in that earlier paper can be improved. Studied in the present paper are rooted non-separable planar maps which may be “nearly cubic”, in the sense that all vertices have degree 3 except possibly the root vertex.

MSC:

05C30 Enumeration in graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
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