  We provide precise asymptotic estimates for the number of several classes of
labelled cubic planar graphs, and we analyze properties of such random graphs
under the uniform distribution. This model was first analyzed by Bodirsky et
al. (Random Structures Algorithms 2007). We revisit their work and obtain new
results on the enumeration of cubic planar graphs and on random cubic planar
graphs. In particular, we determine the exact probability of a random cubic
planar graph being connected, and we show that the distribution of the number
of triangles in random cubic planar graphs is asymptotically normal with linear
expectation and variance. To the best of our knowledge, this is the first time
one is able to determine the asymptotic distribution for the number of copies
of a fixed graph containing a cycle in classes of random planar graphs arising
from planar maps.
