Random cubic planar graphs

We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn−7/2ρ−nn!, where ρ−1 ≐ 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  −ρ 4/4! ≐ 0.999568 and four with probability tending to 1 − e  −ρ 4/4! as n → ∞ with n even. The proof given combines generating function techniques with probabilistic arguments. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

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