Random cubic planar graphs. - Bodirsky, M; Kang, M; Löffler, M; McDiarmid, C (2007)
We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn-7/2 ρ-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-ρ4/4! = 0.999568 and four with probability tending to 1 - e-ρ4/4! as n → ∞ with n even. The proof given combines generating function techniques with probabilistic arguments. © 2006 Wiley Periodicals, Inc.