Skip to main content
SpringerLink
Log in

Bipartite Ramsey Numbers of Cycles for Random Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

For graphs G and H, let \(G\xrightarrow {k}H\) signify that any k-edge coloring of G contains a monochromatic H as a subgraph. Let \(\mathcal{G}(K_2(N),p)\) be random graph spaces with edge probability p, where \(K_2(N)\) is the complete \(N\times N\) bipartite graph. For a constant \(p\in (0,1]\) and a cycle \(C_{2n}\) of length 2n, it is shown that \(\Pr [\mathcal{G}(K_2((k+o(1))n),p)\xrightarrow {k} C_{2n}]\rightarrow 1\) as \(n\rightarrow \infty \), where \(k=2,3\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alon, N., Spencer, J.: The Probabilistic Method. Wiley-Interscience, New York (1992)

    MATH  Google Scholar 

  2. Benevides, F., Skokan, J.: The 3-colored Ramsey number of even cycles. J. Combin. Theory Ser. B 99, 690–708 (2009)

    Article  MathSciNet  Google Scholar 

  3. Bucić, M., Letzter, S., Sudakov, B.: Three colour bipartite Ramsey number of cycles and paths. J. Graph Theory 92, 445–459 (2019)

    Article  MathSciNet  Google Scholar 

  4. Chernoff, H.: A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952)

    Article  MathSciNet  Google Scholar 

  5. Chvátal, V., Rödl, V., Szemerédi, E., Trotter Jr., W.T.: The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B 34, 239–243 (1983)

    Article  MathSciNet  Google Scholar 

  6. Davies, E., Jenssen, M., Roberts, B.: Multicolour ramsey numbers of paths and even cycles. Eur. J. Combin. 63, 124–133 (2017)

    Article  MathSciNet  Google Scholar 

  7. Dudek, A., Prałat, P.: An alternative proof of the linearity of the size Ramsey number of paths. Combin. Probab. Comput. 24, 551–555 (2015)

    Article  MathSciNet  Google Scholar 

  8. Dudek, A., Prałat, P.: On some multicolor Ramsey properties of random graphs. SIAM J. Discrete Math. 31, 2079–2092 (2017)

    Article  MathSciNet  Google Scholar 

  9. Faudree, R., Schelp, R.: All Ramsey numbers for cycles in graphs. Discrete Math. 8, 313–329 (1974)

    Article  MathSciNet  Google Scholar 

  10. Figaj, A., Łuczak, T.: The Ramsey number for a triple of long even cycles. J. Combin. Theory Ser. B 97, 584–596 (2007)

    Article  MathSciNet  Google Scholar 

  11. Figaj, A., Łuczak, T.: The Ramsey number for a triple of long cycles. Combinatorica 38, 827–845 (2018)

    Article  MathSciNet  Google Scholar 

  12. Gerencsér, L., Gyárfás, A.: On Ramsey-type problems. Ann. Univ. Eőtvős Sect. Math. 10, 167–170 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Gyárfás, A., Rousseau, C., Schelp, R.: An extremal problem for paths in bipartite graphs. J. Graph Theory 8, 83–95 (1984)

    Article  MathSciNet  Google Scholar 

  14. Gyárfás, A., Ruszinkó, M., Sárkőzy, G., Szemerédi, E.: Three-color Ramsey numbers for paths. Combinatorica 27, 35–69 (2007)

    Article  MathSciNet  Google Scholar 

  15. Gyárfás, A., Ruszinkó, M., Sárkőzy, G., Szemerédi, E.: Tripartite Ramsey numbers for paths. J. Graph Theory 55, 164–174 (2007)

    Article  MathSciNet  Google Scholar 

  16. Gyárfás, A., Sárkőzy, G.: Star versus two stripes Ramsey numbers and a conjecture of Schelp. Combin. Probab. Comput. 21, 179–186 (2012)

    Article  MathSciNet  Google Scholar 

  17. Janson, S., Łuczak, T., Ruciński, A.: Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)

    Google Scholar 

  18. Jenssen, M., Skokan, J.: Exact Ramsey numbers of odd cycles via nonlinear optimisation. Adv. Math. 376, 107444 (2021)

    Article  MathSciNet  Google Scholar 

  19. Knierim, C., Su, P.: Improved bounds on the multicolor Ramsey numbers of paths and even cycles. Electron. J. Combin. 26, 126 (2019)

    Article  MathSciNet  Google Scholar 

  20. Kohayakawa, Y., Simonovits, M., Skokan, J.: The 3-colored Ramsey number of odd cycles. Electron. Notes Discrete Math. 19, 397–402 (2005)

    Article  MathSciNet  Google Scholar 

  21. Kohayakawa, Y., Rödl, V., Schacht, M., Szemerédi, E.: Sparse partition universal graphs for graphs of bounded degree. Adv. Math. 226, 5041–5065 (2011)

    Article  MathSciNet  Google Scholar 

  22. Letzter, S.: Path Ramsey number for random graphs. Combin. Probab. Comput. 25, 612–622 (2016)

    Article  MathSciNet  Google Scholar 

  23. Łuczak, T.: \(R(C_n, C_n, C_n ) \le (4 + o (1)) n\). J. Combin. Theory Ser. B 75, 174–187 (1999)

    Article  MathSciNet  Google Scholar 

  24. Łuczak, T., Simonovits, M., Skokan, J.: On the multi-colored Ramsey numbers of cycles. J. Graph Theory 69, 169–175 (2012)

    Article  MathSciNet  Google Scholar 

  25. Lin, Q., Li, Y.: A folkman linear family. SIAM J. Discrete Math. 29, 1988–1998 (2015)

    Article  MathSciNet  Google Scholar 

  26. Lin, Q., Li, Y.: Sparse multipartite graphs as partition universal for graphs with bounded degree. J. Comb. Optim. 35, 724–739 (2018)

    Article  MathSciNet  Google Scholar 

  27. Rosta, V.: On a Ramsey-type problem of J. A. Bondy and P. Erdős. J. Combin. Theory Ser. B 15, 94–120 (1973)

    Article  MathSciNet  Google Scholar 

  28. Sárközy, G.: On the multi-colored Ramsey numbers of paths and even cycles. Electron. J. Combin. 23, 3.53 (2016)

    Article  MathSciNet  Google Scholar 

  29. Shen, L., Lin, Q., Liu, Q.: Bipartite Ramsey numbers for bipartite graphs of small bandwidth. Electron. J. Combin. 25, 2.16 (2018)

    Article  MathSciNet  Google Scholar 

  30. Szemerédi, E.: Regular partitions of graphs, Problémes combinatories et théorie des graphs, Orsay 1976, J.C. Bermond, J.C. Fournier, M. Las Vergnas, D. Scotteau, (Editors), Colloq. Internat. CNRS 260, Paris (1978)

Download references

Acknowledgements

We are grateful to the Managing Editor Professor Ingo Schiermeyer and reviewers for their invaluable comments and suggestions which improve the presentations of the results greatly. Particularly, one of reviewers pointed that we should use the standard regularity lemma instead of its sparse form in the original manuscript that was artificially more complicated, he/she also pointed that Claim 1 on Page 4 can be proved by an easy application of the union bound instead of Markov’s inequality.

Funding

Supported in part by NSFC (11871377, 11901001, 11931002).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China

    Meng Liu

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Yusheng Li

Authors
  1. Meng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yusheng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meng Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, M., Li, Y. Bipartite Ramsey Numbers of Cycles for Random Graphs. Graphs and Combinatorics 37, 2703–2711 (2021). https://doi.org/10.1007/s00373-021-02386-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-021-02386-7

Keywords

Search

Navigation