Local Convergence and Stability of Tight Bridge-Addable Graph Classes
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.
bridge-addable classes
random graphs
stability
local convergence
random forests
26:1-26:11
Regular Paper
Guillaume
Chapuy
Guillaume Chapuy
Guillem
Perarnau
Guillem Perarnau
10.4230/LIPIcs.APPROX-RANDOM.2016.26
Louigi Addario-Berry, Colin McDiarmid, and Bruce Reed. Connectivity for bridge-addable monotone graph classes. Combin. Probab. Comput., 21(6):803-815, 2012.
Paul Balister, Béla Bollobás, and Stefanie Gerke. Connectivity of addable graph classes. J. Combin. Theory Ser. B, 98(3):577-584, 2008.
Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001.
Guillaume Chapuy and Guillem Perarnau. Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture. Extended abstract in the proceedings of SODA 2016. Long version submitted for publication, see arXiv:1504.06344., 2015.
Guillaume Chapuy and Guillem Perarnau. Local convergence and stability of tight bridge-addable graph classes. In preparation., 2016.
Paul Erdős. On some new inequalities concerning extremal properties of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 77-81, 1966.
Paul Erdős. Some recent results on extremal problems in graph theory. Results, Theory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, New York, pages 117-123, 1967.
Paul Erdős and M Simonovits. A limit theorem in graph theory. In Studia Sci. Math. Hung. Citeseer, 1966.
Mihyun Kang and Konstantinos Panagiotou. On the connectivity of random graphs from addable classes. J. Combin. Theory Ser. B, 103(2):306-312, 2013.
László Lovász. Large networks and graph limits, volume 60 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2012.
Colin McDiarmid, Angelika Steger, and Dominic J. A. Welsh. Random graphs from planar and other addable classes. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 231-246. Springer, Berlin, 2006.
Alfréd Rényi. Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl., 4:73-85, 1959.
Miklós Simonovits. A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279-319, 1968.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode