Random planar graphs

Random planar graphs

We study various properties of the random planar graph Rn, drawn uniformly at random from the classPn of all simple planar graphs on n labelled vertices. In particular, we show that the probability thatRn is connected is bounded away from 0 and from 1.We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding R...

Random cubic planar graphs

We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αnρn!, where ρ . = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e /4! . = 0.999568, and is four with probab...

Random planar graphs and the number of planar graphs

We survey recent progress on the enumeration of labelled planar graph and on the distribution of several parameters in random planar graphs. The main goal is to show how to use generating functions to obtain asymptotic estimates and limit probability laws. The emphasis is on presenting the main ideas while keeping technical details to a minimum.

Random cubic planar graphs revisited

Random planar graphs and beyond

We survey several results on the enumeration of planar graphs and on properties of random planar graphs. This includes basic parameters, such as the number of edges and the number of connected components, and extremal parameters such as the size of the largest component, the diameter and the maximum degree. We discuss extensions to graphs on surfaces and to classes of graphs closed under minors...