License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.26
URN: urn:nbn:de:0030-drops-66494
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6649/
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Chapuy, Guillaume ; Perarnau, Guillem

Local Convergence and Stability of Tight Bridge-Addable Graph Classes

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Abstract

A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests.

In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class.

BibTeX - Entry

@InProceedings{chapuy_et_al:LIPIcs:2016:6649,
  author =	{Guillaume Chapuy and Guillem Perarnau},
  title =	{{Local Convergence and Stability of Tight Bridge-Addable Graph Classes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{26:1--26:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6649},
  URN =		{urn:nbn:de:0030-drops-66494},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.26},
  annote =	{Keywords: bridge-addable classes, random graphs, stability, local convergence, random forests}
}

Keywords: bridge-addable classes, random graphs, stability, local convergence, random forests
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
Issue Date: 2016
Date of publication: 06.09.2016

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