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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.

Random Planar Maps and Graphs with Minimum Degree Two and Three