Abstract
We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erdős–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erdős–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log2 n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.
A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.
A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 1, 107–146 (2001) [RussianMath. Surveys 56 1, 103–139 (2001)].
A. M. Raigorodskii, “On the chromatic numbers of spheres in Euclidean spaces,” Dokl. Ross. Akad. Nauk 432 2, 174–177 (2010) [Dokl. Math. 81 3, 379–382 (2010)].
A. M. Raigorodskii, “On the chromatic numbers of spheres in Rn,” Combinatorica 32 1, 111–123 (2012).
J. Balogh, A. Kostochka, and A. Raigorodskii, “Coloring some finite sets in Rn,” Discuss. Math. Graph Theory 33 1, 25–31 (2013).
J. Pach and P. K. Agarwal, Combinatorial Geometry (John Wiley and Sons, New York, 1995).
L. A. Székely, “Erdős on unit distances and the Szemerédi–Trotter theorems,” in Paul Erdős and HisMathematics, II, Bolyai Soc. Math. Stud. (János BolyaiMath. Soc., Budapest, 2002), Vol. 11, pp. 649–666.
A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of Its Creators (Springer, New York, 2009).
V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, in The DolcianiMath. Exp. (Math. Assoc. America, Washington, DC, 1991), Vol. 11.
V. Boltyanski, H. Martini, and P. S. Soltan, Excursions into Combinatorial Geometry, in Universitext (Springer-Verlag, Berlin, 1997).
A. M. Raigorodskii, “Three lectures on the Borsuk partition problem,” in Surveys in Contemporary Mathematics, LondonMath. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2008), Vol. 347, pp. 202–248.
A. M. Raigorodskii, “Around Borsuk’s hypothesis,” Sovrem. Mat. Fundam. Napravl. 23, 147–164 (2007) [J. Math. Sci. (New York) 154 4, 604–623 (2008)].
R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory (JohnWiley and Sons, New York, 1990).
Z. Nagy, “A certain constructive estimate of the Ramsey number,” Mat. Lapok 23, 301–302 (1972).
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Parts I, II (North-Holland, Amsterdam–New York–Oxford, 1977; Radio i Svyaz’, Moscow, 1979).
L. Bassalygo, G. Cohen, and G. Zémor, “Codes with forbidden distances,” DiscreteMath. 213, 3–11 (2000).
B. Bollobás, Random Graphs, in Cambridge Stud. in Adv. Math. (Cambridge Univ. Press, Cambridge, 2001), Vol. 73.
S. Janson, T. Luczak, and A. Rucinski, Random Graphs (Wiley, New York, 2000).
V. F. Kolchin, Random Graphs (Fizmatlit, Moscow, 2000) [in Russian].
L. I. Bogolyubskii, A S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of random subgraphs in some sequences of graphs,” Dokl. Ross. Akad. Nauk 457 4, 383–387 (2014) [Dokl. Math. 90 1, 462–465 (2014)].
L. I. Bogolyubskii, A S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of the random subgraphs of some distance graphs,” Mat. Sb. 206 10, 3–36 (2015) [Sb. Math. 206 10, 1340–1374 (2015)].
M. M. Pyaderkin, “Independence numbers of random subgraphs of a distance graph,” Mat. Zametki 99 2, 288–297 (2016) [Math. Notes 99 (1–2), 312–319 (2016)].
M. M. Pyaderkin, “On the stability of the Erdős–Ko–Rado theorem,” Dokl. Ross. Akad. Nauk 462 2, 144–147 (2015) [Dokl. Math. 91 3, 290–293 (2015)].
B. Bollobás, B. P. Narayanan, and A. M. Raigorodskii, “On the stability of the Erdős–Ko–Rado theorem,” J. Combin. Theory Ser. A 137, 64–78 (2016).
A. M. Raigorodskii, “Combinatorial geometry and coding theory,” Fundam. Inform. (2016) (in press).
P. Frankl and Z. Füredi, “Forbidding just one intersection,” J. Combin. Theory Ser. A 39 2, 160–176 (1985).
J. Matousek, Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, in Universitext (Springer-Verlag, Berlin, 2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M. M. Pyaderkin, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 4, pp. 564–573.
Rights and permissions
About this article
Cite this article
Pyaderkin, M.M. Independence numbers of random subgraphs of distance graphs. Math Notes 99, 556–563 (2016). https://doi.org/10.1134/S0001434616030299
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616030299