Mathematics – Combinatorics
Scientific paper
2010-06-02
Mathematics
Combinatorics
30 pages, 1 figure
Scientific paper
Let $P(n,M)$ be a graph chosen uniformly at random from the family of all labeled planar graphs with $n$ vertices and $M$ edges. In the paper we study the component structure of $P(n,M)$. Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of $P(n,M)$. The first one, of width $\Theta(n^{2/3})$, is analogous to the phase transition observed in the standard random graph models and takes place for $M=n/2+O(n^{2/3})$, when the largest complex component is formed. Then, for $M=n+O(n^{3/5})$, when the complex components cover nearly all vertices, the second critical period of width $n^{3/5}$ occurs. Starting from that moment increasing of $M$ mostly affects the density of the complex components, not its size.
Kang Mihyun
Łuczak Tomasz
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