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On the diameter of random planar graphs

Abstract : We show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs.
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Guillaume Chapuy, Eric Fusy, Omer Gimenez, Marc Noy. On the diameter of random planar graphs. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.65-78. ⟨hal-00714713v2⟩

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