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} wll contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n tends to infinity.Comment: 24 pages, 12 figure

Characterisation of symmetries of unlabelled triangulations and its applications

We give a full characterisation of the symmetries of unlabelled triangulations and derive a constructive decomposition of unlabelled triangulations depending on their symmetries. As an application of these results we can deduce a complete enumerative description of unlabelled cubic planar graphs.Comment: 37 pages, 21 figures. An extended abstract of this paper has been accepted for the proceedings of EUROCOMB 201