The maximum degree of random planar graphs

Abstract

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.) c1logn<normalΔn<c2logn, for suitable constants . In this paper, we determine the precise asymptotics, showing in particular that w.h.p. |normalΔn−cprefixlogn|=O(loglogn), for a constant c≈2.52946 that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.

On the Diameter of Random Planar Graphs

“…It is still open and seems technically very involved to show a limit distribution for the profile or radius of a random connected planar graph rescaled by n 1/4 . Other extremal parameters that have been analysed recently in random planar graphs using analytic techniques are the size of the largest k-connected component [22,30] and the maximum vertex degree [12,13].…”

Concentration of maximum degree in random planar graphs

“…They proved that whp ∆ (P (n)) = Θ log n . Later Drmota, Giménez, Noy, Panagiotou, and Steger [13] used tools from analytic combinatorics and Boltzmann sampling techniques to show that whp ∆ (P (n)) is concentrated in an interval of length O log log n .…”

“…a vertex with degree one, with largest label and thereby build a sequence by noting the unique neighbours of the leaves. We will show in Theorem 6.1 that this is indeed a bijection and that the degree of a vertex v is determined by the number of occurrences of v in the sequence (see (13)). It is straightforward to construct a random element from S (n, t ) by a balls-into-bins model such that the load of a bin equals the number of occurrences in the sequence of the corresponding element.…”

“…To show (ii), we suppose that we first apply the algorithm to obtain ψ(F ) and then the algorithm ψ −1 with input w = ψ(F ). Due to (13) the degree sequence of F 0 = F equals d0 (1), . .…”

On the diameter of random planar graphs

“…It is still open and seems technically very involved to show a limit distribution for the profile or radius of a random connected planar graph rescaled by n 1/4 . Other extremal parameters that have been analyzed recently in random planar graphs using analytic techniques are the size of the largest k-connected component [22,30] and the maximum vertex degree [12,13].…”