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Random graphs on surfaces

Authors
Journal
Journal of Combinatorial Theory Series B
0095-8956
Publisher
Elsevier
Publication Date
Volume
98
Issue
4
Identifiers
DOI: 10.1016/j.jctb.2007.11.006
Keywords
  • Random Graph
  • Surface
  • Planar
  • Embeddable
  • Enumeration
  • Labelled

Abstract

Abstract Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set { 1 , … , n } , then with probability tending to 1 as n → ∞ , this random graph either is connected or consists of one giant component together with a few nodes in small planar components.

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