Abstract
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.
Highlights
The main goal of this paper is to enumerate planar graphs subject to a condition on the minimum degree δ, and to analyze the corresponding random planar graphs
We need the following result from [6]. It deals with the maximum degree of random graphs and can be adapted to other extremal parameters such as the size of the largest tree attached to the core
Interesting examples are the classes of series-parallel and outerplanar graphs
Summary
The main goal of this paper is to enumerate planar graphs subject to a condition on the minimum degree δ, and to analyze the corresponding random planar graphs. Analyzing the bivariate generating function C(x, y) it is possible to obtain results on the number of edges and other basic parameters in random planar graphs. We show that the corresponding constants for planar 2-graphs and 3-graphs are μ2 ≈ 2.2614, μ3 ≈ 2.4065 This conforms to our intuition that increasing the minimum degree should increase the expected number of edges. We analyze the size Xn of the core in a random connected planar graph, and the size Yn of the kernel in a random planar 2-graph We show that both variables are asymptotically normal with linear expectation and variance and that. We remark that the expected size of the largest block (2-connected component) in random connected planar graphs is asymptotically 0.9598n [12] This is consistent since the largest block is contained in the core but not .
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