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The Turán number of book graphs

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Abstract

Given a graph H and a positive integer n,  the Turán number of H for the order n,  denoted by \(\textrm{ex}(n,H),\) is the maximum size of a simple graph of order n not containing H as a subgraph. The book with p pages, denoted by \(B_p\), is the graph that consists of p triangles sharing a common edge. Bollobás and Erdős initiated the research on the Turán number of book graphs in 1975. The two numbers \(\textrm{ex}(p+2,B_p)\) and \(\textrm{ex}(p+3,B_p)\) have been determined by Qiao and Zhan. In this paper we determine the numbers \(\textrm{ex}(p+4,B_p),\) \(\textrm{ex}(p+5,B_p)\) and \(\textrm{ex}(p+6,B_p),\) and characterize the corresponding extremal graphs for the numbers \(\textrm{ex}(n,B_p)\) with \(n=p+2,\,p+3,\,p+4,\,p+5.\)

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References

  1. B. Bollobás, Extremal Graph Theory, Academic Press, London-New York, 1978.

    MATH  Google Scholar 

  2. B. Bollobás and P. Erdős, Unsolved problems, in Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), pp.678-696, Utilitas Math., Winnipeg, Man., 1976.

  3. C.S. Edwards, A lower bound for the largest number of triangles with a common edge (unpublished manuscript), 1977.

  4. W. Mantel, Problem 28, Wiskundige Opgaven, 10(1907), 60-61.

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  5. P. Qiao and X. Zhan, On a problem of Erdős about graphs whose size is the Turán number plus one, Bull. Aust. Math. Soc., 105(2022), 177-187.

    Article  MathSciNet  MATH  Google Scholar 

  6. D.B. West, Introduction to Graph Theory, Prentice Hall, Inc., 1996.

    MATH  Google Scholar 

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Acknowledgements

This research was supported by the NSFC grant 12271170 and Science and Technology Commission of Shanghai Municipality grant 22DZ2229014.

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Jingru Yan & Xingzhi Zhan

Authors
  1. Jingru Yan
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  2. Xingzhi Zhan
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Correspondence to Xingzhi Zhan.

Additional information

Communicated by Shariefuddin Pirzada.

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Yan, J., Zhan, X. The Turán number of book graphs. Indian J Pure Appl Math (2023). https://doi.org/10.1007/s13226-023-00467-2

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  • DOI: https://doi.org/10.1007/s13226-023-00467-2

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