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Local Convergence and Stability of Tight Bridge-addable Classes

  • G. Chapuy (a1) and G. Perarnau (a1)


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 ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$ -vertex graph from  ${\mathcal{G}}$ , then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$ . The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.

In this paper, we prove a form of uniqueness in this statement: if ${\mathcal{G}}$ 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 $n$ -vertex random forest in a local sense. For example, if the probability converges to  $e^{-1/2}$ , then $G_{n}$ converges in the sense of Benjamini–Schramm to the uniformly infinite random forest  $F_{\infty }$ . This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.



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Supported by the European Research Council, grant ERC-2016-STG 716083 “CombiTop”.



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[ABMR12] Addario-Berry, L., McDiarmid, C., and Reed, B., Connectivity for bridge-addable monotone graph classes . Combin. Probab. Comput. 21(2012), 803815.
[Ald93] Aldous, D., The continuum random tree. III . Ann. Probab. 23(1993), 248289.
[Ald98] Aldous, D., Tree-valued Markov chains and Poisson-Galton-Watson distributions . In: Microsurveys in discrete probability (Princeton, NJ, 1997), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Amer. Math. Soc., Providence, RI, 199, pp. 120.
[BBG08] Balister, P., Bollobás, B., and Gerke, S., Connectivity of addable graph classes . J. Combin. Theory Ser. B 98(2008), 577584.
[BLL98] Bergeron, F., Labelle, G., and Leroux, P., Combinatorial species and tree-like structures. Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998.
[BS01] Benjamini, I. and Schramm, O., Recurrence of distributional limits of finite planar graphs . Electron. J. Probab. 6(2001), no. 23.
[CP15] Chapuy, G. and Perarnau, G., Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture . J. Combin. Theory Ser. B.
[Erd66] Erdős, P., On some new inequalities concerning extremal properties of graphs . In: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1966, pp. 7781.
[Erd67] Erdős, P., Some recent results on extremal problems in graph theory . In: Results, Theory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, New York, 1967, pp. 117123.
[ES66] Erdős, P. and Simonovits, M., A limit theorem in graph theory . Studia Sci. Math. Hungar 1(1966), 5157.
[FS09] Flajolet, P. and Sedgewick, R., Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
[KP13] Kang, M. and Panagiotou, K., On the connectivity of random graphs from addable classes . J. Combin. Theory Ser. B 103(2013), 306312.
[Lov12] Lovász, L., Large networks and graph limits. American Mathematical Society Colloquium Publications, 60, American Mathematical Society, Providence, RI, 2012.
[MSW05] McDiarmid, C., Steger, A., and Welsh, D. J. A., Random planar graphs . J. Combin. Theory Ser. B 93(2005), 187205.
[MSW06] McDiarmid, C., Steger, A., and Welsh, D. J. A., Random graphs from planar and other addable classes . In: Topics in discrete mathematics, Algorithms Combin., 26, Springer, Berlin, pp. 231246.
[Rén59] Rényi, A., Some remarks on the theory of trees . Magyar Tud. Akad. Mat. Kutató Int. Közl. 4(1959), 7385.
[Sim68] Simonovits, M., A method for solving extremal problems in graph theory, stability problems . In: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, pp. 279319.
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Local Convergence and Stability of Tight Bridge-addable Classes

  • G. Chapuy (a1) and G. Perarnau (a1)


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