Building on previous work by the present authors [Proc. London Math. Soc.
119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number
gn of labelled 4-regular planar graphs. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the
enumeration of planar graphs. In addition to analytic methods, our solution
needs intensive use of computer algebra in order to work with large systems of
polynomials equations. In particular, we use evaluation and Lagrange
interpolation in order to compute resultants of multivariate polynomials. We
also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular
planar graphs, and for the number of 4-regular simple maps, both connected and
2-connected.Comment: 23 pages, including 5 pages of appendix. Corrected titl