Random planar maps and graphs with minimum degree two and three
  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. 
Marc Noy, Lander Ramos
math.CO math.PR