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On the Turán Number of Theta Graphs

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Abstract

The Turán number ex(nH) is the maximum number of edges in any graph of order n that contains no copy of H as a subgraph. For any three positive integers pqr with \(p\le q\le r\) and \(q\ge 2\), let \(\theta (p,q,r)\) denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths pqr, respectively. Let \(k=p+q+r-1\). In this paper, we obtain the exact value of \(ex(n,\theta (p,q,r))\) and characterize the unique extremal graph for \(n\ge 9k^2-3k\) and any pqr with different parities. This extends a known result on odd cycles.

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References

  1. Bondy, J.A., Simonovits, M.: Cycles of even length in graphs. J. Comb. Theory Ser. B 16, 97–105 (1974)

    Article  MathSciNet  Google Scholar 

  2. Bukh, B., Tait, M.: Turán numbers of theta graphs. Comb. Probab. Comput. 29, 495–507 (2020)

    Article  Google Scholar 

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

    Article  Google Scholar 

  4. Dowden, C.: Extremal \(C_4\)-free / \(C_5\)-free planar graphs. J. Graph Theory 83, 213–230 (2016)

    Article  MathSciNet  Google Scholar 

  5. Dzido, T.: A note on Turán numbers for even wheels. Graphs Comb. 29, 1305–1309 (2013)

    Article  Google Scholar 

  6. Dzido, T., Kubale, M., Piwakowski, K.: On some Ramsey and Turán-type numbers for paths and cycles. Electron. J. Comb. 13(1), 9 (2006). (Research paper 55)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Faudree, R.J., Schelp, R.H.: Path Ramsey numbers in multicolorings. J. Comb. Theory Ser. B 19, 150–160 (1975)

    Article  MathSciNet  Google Scholar 

  9. Faudree, R.J., Simonovits, M.: On a class of degenerate extremal graph problems. Combinatorica 3(1), 83–93 (1983)

    Article  MathSciNet  Google Scholar 

  10. Füredi, Z., Gunderson, D.S.: Extremal numbers for odd cycles. Comb. Probab. Comput. 24, 641–645 (2015)

    Article  MathSciNet  Google Scholar 

  11. Kopylov, G.N.: Maximal paths and cycles in a graph. Dokl. Akad. Nauk SSSR 234, 19–21 (1977)

    MathSciNet  Google Scholar 

  12. Mantel, W.: Problem 28, soln. by Gouventak, H., Mantel, W., Teixeira de Mattes, J., Schuh, F. and Wythoff, W.A. Wiskundige Opgaven. 10, 60–61 (1907)

  13. Lan, Y.X., Shi, Y.T., Song, Z.X.: Extremal theta-free planar graphs. Discret. Math. 342, 111610 (2019)

    Article  MathSciNet  Google Scholar 

  14. Lan, Y.X., Shi, Y.T., Tu, J.H.: The Turán number of star forests. Appl. Math. Comput. 348, 2701–274 (2019)

    MATH  Google Scholar 

  15. Lidický, B., Liu, H., Palmer, C.: On the Turán number of forests. Electron. J. Comb. 20(2), 13 (2013). (Research paper 62)

    Article  Google Scholar 

  16. Simonovits, M.: A method for solving extremal problems in graph theory, stability problems. In: Theory of Graphs. Academic Press, New York (1968)

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

    MathSciNet  Google Scholar 

  18. Verstraëte, J., Williford, J.: Graphs without theta subgraphs. J. Comb. Theory Ser. B 134, 76–87 (2019)

    Article  MathSciNet  Google Scholar 

  19. Wang, J., Yang, W.H.: The Turán number for spanning linear forests. Discret. Appl. Math. 254, 291–294 (2019)

    Article  Google Scholar 

  20. Yuan, L.T., Zhang, X.D.: The Turán number of disjoint copies of paths. Discret. Math. 340, 132–139 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions. The research of Mingqing Zhai is supported by NSFC (No. 11971445).

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Correspondence to Jinlong Shu.

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Zhai, M., Fang, L. & Shu, J. On the Turán Number of Theta Graphs. Graphs and Combinatorics 37, 2155–2165 (2021). https://doi.org/10.1007/s00373-021-02342-5

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  • DOI: https://doi.org/10.1007/s00373-021-02342-5

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