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Clique Numbers of Random Subgraphs of Some Distance Graphs

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Abstract

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

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References

  1. Brass, P., Moser, W., and Pach, J., Research Problems in Discrete Geometry, New York: Springer, 2005.

    MATH  Google Scholar 

  2. Raigorodskii, A.M., Coloring Distance Graphs and Graphs of Diameters, Thirty Essays on Geometric Graph Theory, Pach, J., Ed., New York: Springer, 2013, pp. 429–460.

    Book  MATH  Google Scholar 

  3. Raigorodskii, A.M., Cliques and Cycles in Distance Graphs and Graphs of Diameters, Discrete Geometry and Algebraic Combinatorics, Barg, A. and Musin, O.R., Eds., Providence, RI: Amer. Math. Soc., 2014, pp. 93–109.

    MATH  Google Scholar 

  4. Raigorodskii, A.M., Combinatorial Geometry and Coding Theory, Fund. Inform., 2016, vol. 145, no. 3, pp. 359–369.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bobu, A.V., Kupriyanov, A.E., and Raigorodskii, A.M., On the Chromatic Numbers of Distance Graphs That Are Similar to Kneser Graphs, Dokl. Akad. Nauk, 2016, vol. 468, no. 3, pp. 247–250 [Dokl. Math. (Engl. Transl.), 2016, vol. 93, no. 3, pp. 267–269].

    MathSciNet  MATH  Google Scholar 

  6. Bogolyubskii, L.I., Gusev, A.S., Pyaderkin, M.M., and Raigorodskii, A.M., The Independence Numbers and Chromatic Numbers of Random Subgraphs in Some Sequences of Graphs, Dokl. Akad. Nauk, 2014, vol. 457, no. 4, pp. 383–387 [Dokl. Math. (Engl. Transl.), 2014, vol. 90, no. 1, pp. 462–465].

    MathSciNet  MATH  Google Scholar 

  7. Bogolyubskii, L.I., Gusev, A.S., Pyaderkin, M.M., and Raigorodskii, A.M., Independence Numbers and Chromatic Numbers of Random Subgraphs in Some Sequences of Graphs, Mat. Sb., 2015, vol. 206, no. 10, pp. 3–36 [Sb. Math. (Engl. Transl.), 2015, vol. 206, no. 9–10, pp. 1340–1374].

    Article  MATH  Google Scholar 

  8. Gusev, A.S., New Upper Bound for the Chromatic Number of a Random Subgraph of a Distance Graph, Mat. Zametki, 2015, vol. 97, no. 3, pp. 342–349 [Math. Notes (Engl. Transl.), 2015, vol. 97, no. 3–4, pp. 326–332].

    Article  MathSciNet  MATH  Google Scholar 

  9. Bollobás, B., The Chromatic Number of Random Graphs, Combinatorica, 1988, vol. 8, no. 1, pp. 49–55.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bollobás, B., Narayanan, B.P., and Raigorodskii, A.M., On the Stability of the Erdős–Ko–Rado Theorem, J. Combin. Theory Ser. A, 2016, vol. 137, pp. 64–78.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kupavskii, A., On Random Subgraphs of Kneser and Schrijver Graphs, J. Combin. Theory Ser. A, 2016, vol. 141, pp. 8–15.

    Article  MathSciNet  MATH  Google Scholar 

  12. Pyaderkin, M.M., Independence Numbers of Random Subgraphs of Distance Graphs, Mat. Zametki, 2016, vol. 99, no. 4, pp. 564–573 [Math. Notes (Engl. Transl.), 2016, vol. 99, no. 3–4, pp. 556–563].

    Article  MathSciNet  MATH  Google Scholar 

  13. Pyaderkin, M.M. and Raigorodskii, A.M., On Random Subgraphs of Kneser Graphs and Their Generalizations, Dokl. Akad. Nauk, 2016, vol. 470, no. 4, pp. 384–386 [Dokl. Math. (Engl. Transl.), 2016, vol. 94, no. 2, pp. 547–549].

    MATH  Google Scholar 

  14. Pyaderkin, M.M., On the Stability of Some Erdős–Ko–Rado Type Results, Discrete Math., 2017, vol. 340, no. 4, pp. 822–831.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Statistics and Random Processes, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    A. S. Gusev

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  1. A. S. Gusev
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Correspondence to A. S. Gusev.

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Original Russian Text © A.S. Gusev, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 2, pp. 73–85.

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Gusev, A.S. Clique Numbers of Random Subgraphs of Some Distance Graphs. Probl Inf Transm 54, 165–175 (2018). https://doi.org/10.1134/S0032946018020059

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  • DOI: https://doi.org/10.1134/S0032946018020059

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