We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.
Noy, M.; Ramos, L. Random planar maps and graphs with minimum degree two and three. "Electronic journal of combinatorics", 5 Octubre 2018, vol. 25, núm. 4, p. 1-38.