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Generalized Turán Number of Even Linear Forests

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Abstract

The generalized Turán number \(ex(n,K_s,H)\) is defined to be the maximum number of copies of a complete graph \(K_s\) in any H-free graph on n vertices. Let F be an even linear forest, that is, F consists of paths of even order. In this paper, we determine the value of \(ex(n,K_s,F)\), which generalizes or improves some known results on generalized Turán number and classical Turán number.

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References

  1. Alon, N., Kostochka, A., Shikhelman, C.: Many \(T\) copies in \(H\)-free subgraphs of random graphs. J. Comb. 9, 567–597 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Alon, N., Shikhelman, C.: Many \(T\) copies in \(H\)-free graphs. J. Comb. Theory Ser. B 121, 146–172 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bollobás, B., Győri, E.: Pentagons vs. triangles. Discret. Math. 308, 4332–4336 (2008)

    Article  MathSciNet  Google Scholar 

  4. Bushaw, N., Kettle, N.: Turán numbers of multiple paths and equibipartite forests. Comb. Probab. Comput. 20, 837–853 (2011)

    Article  Google Scholar 

  5. Chase, Z.: A proof of the Gan-Loh-Sudakov conjecture (2019) arXiv:1912.01600

  6. Cutler, J., Radcliffe, A.J.: The maximum number of complete subgraphs of fixed size in a graph with given maximum degree. J. Graph Theory 84, 134–145 (2017)

    Article  MathSciNet  Google Scholar 

  7. Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)

    Article  MathSciNet  Google Scholar 

  8. Erdős, P.: On the number of complete subgraphs contained in certain graphs. Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 7, 459–474 (1962)

    MathSciNet  MATH  Google Scholar 

  9. Erdős, P., Gallai, T.: On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hung. 10, 337–356 (1959)

    Article  MathSciNet  Google Scholar 

  10. Gerbner, D., Győri, E., Methuku, A., Vizer, M.: Generalized Turán problems for even cycles. J. Comb. Theory Ser. B 145, 169–213 (2020)

    Article  Google Scholar 

  11. Gerbner, D., Keszegh, B., Palmer, C., Patkós, B.: On the number of cycles in a graph with restricted cycle lengths. SIAM J. Discret. Math. 32, 266–279 (2018)

    Article  MathSciNet  Google Scholar 

  12. Gerbner, D., Methuku, A., Vizer, M.: Generalized Turán problems for disjoint copies of graphs. Discret. Math. 342, 3130–3141 (2019)

    Article  Google Scholar 

  13. Gerbner, D., Palmer, C.: Counting copies of a fixed subgraph in \(F\)-free graphs. Eur. J. Comb. 82, 103001 (2019). (15 pages)

    Article  MathSciNet  Google Scholar 

  14. Grzesik, A.: On the maximum number of five-cycles in a triangle-free graph. J. Comb. Theory Ser. B 102, 1061–1066 (2012)

    Article  MathSciNet  Google Scholar 

  15. Győri, E., Li, H.: The maximum number of triangles in \(C_{2k+1}\)-free graph. Comb. Probab. Comput. 21, 187–191 (2012)

    Article  Google Scholar 

  16. Hatami, H., Hladký, J., Král, D., Norine, S., Razborov, A.: On the number of Pentagons in triangle-free graphs. J. Comb. Theory Ser. A 120, 722–732 (2013)

    Article  MathSciNet  Google Scholar 

  17. Khormali, O.: Extremal problems for forests in graphs and hypergraphs, Graduate Student Theses, Dissertations, Professional Papers. 11483 (2019) https://scholarworks.umt.edu/etd/11483

  18. Lidický, B., Liu, H., Palmer, C.: On the Turán number of forests, Electron. J. Combin. 20, Paper 62, 13 pages (2013)

  19. Luo, R.: The maximum number of cliques in graphs without long cycles. J. Comb. Theory Ser. B 128, 219–226 (2017)

    Article  MathSciNet  Google Scholar 

  20. Ma, J., Qiu, Y.: Some sharp results on the generalized Turán numbers. Eur. J. Comb. 84, 103026 (2020). (16 pages)

    Article  Google Scholar 

  21. Ning, B., Peng, X.: Extensions of the Erdős-Gallai theorem and Luo’s theorem. Comb. Probab. Comput. 29, 128–136 (2020)

    Article  Google Scholar 

  22. Turán, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452 (1941)

    MathSciNet  Google Scholar 

  23. Wang, J.: The shifting method and generalized Turán number of matchings. Eur. J. Comb. 85, 103057 (2020). (7 pages)

    Article  Google Scholar 

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Acknowledgements

We are grateful to the anonymous referees for their very careful comments. This research was supported by NSFC under Grant Nos. 11871270 and 11931006.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xiutao Zhu & Yaojun Chen

  2. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, People’s Republic of China

    Fangfang Zhang

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  1. Xiutao Zhu
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  2. Fangfang Zhang
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Correspondence to Yaojun Chen.

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Zhu, X., Zhang, F. & Chen, Y. Generalized Turán Number of Even Linear Forests. Graphs and Combinatorics 37, 1437–1449 (2021). https://doi.org/10.1007/s00373-021-02329-2

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