Random Planar Graphs with Bounds on the Maximum and Minimum Degrees

Let P _n,d,D denote the graph taken uniformly at random from the set of all labelled planar graphs on {1, 2, . . . , n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability that P _n,d,D will contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n → ∞.