The maximum degree of random planar graphs
نویسندگان
چکیده
McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δn of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c1 logn < Δn < c2 logn, for suitable constants 0 < c1 < c2. In this paper, we determine the precise asymptotics, showing in particular that w.h.p. |Δn − c logn| = O(log log n), for a constant c ≈ 2.52946 that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.
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