The number of rooted nearly cubic \(c\)-nets (Q5931896)

A \(c\)-net is a map whose underlying graph is 3-connected, having more than 3 edges. From the authors' introduction: ``The purpose of this paper is, firstly, to obtain a parametric expression of the enumerating function for rooted nearly cubic \(c\)-nets with the size and the root-vertex valency of the maps as two parameters, via nonseparable nearly cubic planar maps \(\dots\). On this basis \(\dots\) explicit expressions of the enumerating functions can be derived immediately by employing Lagrangian inversion.'' By \({\mathcal M}_{nc}\) and \({\mathcal M}_{ns}\) the authors denote, respectively, the sets of rooted nearly cubic planar \(c\)-nets and rooted nearly cubic planar nonseparable maps. \S 2 is concerned with relations between the enumerating functions for \({\mathcal M}_{nc}\) and \({\mathcal M}_{ns}\); in \S 3 these relations are exploited to deduce parametric expressions for the enumerator for \({\mathcal M}_{nc}\); explicit formulae for the numbers are developed in \S 4. The derivation follows the methods developed by \textit{W. T. Tutte} [e.g., A census of planar maps, Can. J. Math. 15, 249-271 (1963; Zbl 0115.17305)], and \textit{R. C. Mullin} and \textit{P. J. Schellenberg} [The enumeration of \(c\)-nets via quadrangulations, J. Comb. Theory 4, 259-276 (1968; Zbl 0183.52403)], and the reviewer [On the existence of square roots in certain rings of power series, Math. Ann. 158, 82-89 (1965; Zbl 0136.02503)]. (The bibliographic reference given by the authors for the last mentioned paper is incorrect; they have given the data for another, related, paper of the reviewer and \textit{W. T. Tutte} [On the enumeration of rooted non-separable planar maps, Can. J. Math. 16, 572-577 (1964; Zbl 0119.38804)]).