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Planar Turán Numbers of Short Paths

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Abstract

Given a graph H, the planar Turán number of H, denoted \(ex_{\mathcal {P}}(n,H),\) is the maximum number of edges in an H-free planar graph on n vertices. The idea of determining \(ex_{\mathcal {P}}(n,P_k)\) was promoted by Lan, Song and Shi, in which they obtained that the planar Turán number of paths \(P_k\) with \(k\in \{8,9\}\). In this paper, we determine the planar Turán number of paths \(P_k\) with \(k\in \{6,7,10,11\}\).

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References

  1. Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Connected graphs without long paths. Discrete Math. 308, 4487–4494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bollobás, B.: Modern Graph Theory. Springer, New York (2013)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdős, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2, 275–288 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Erdős, P., Stone, A.H.: On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091 (1946)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Füredi, Z.: Turán type problems. In: Surveys in Combinatorics, London Mathematical Society Lecture Note Series 166, pp. 253–300. Cambridge University Press, Cambridge (1991)

  9. Füredi, Z., Jiang, T.: Hypergraph Turán numbers of linear cycles. J. Comb. Theory Ser. A 123, 252–270 (2014)

    Article  MATH  Google Scholar 

  10. Füredi, Z., Jiang, T., Seiver, R.: Exact solution of the hypergraph Turán problem for \(k\)-uniform linear paths. Combinatorica 34, 299–322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keevash, P.: Hypergraph Turán problems. In: Surveys in Combinatorics 2011, London Mathematical Society Lecture Note Series 392, pp. 83–139. Cambridge University Press, Cambridge (2011)

  12. Kostochka, A., Mubayi, D., Verstraëte, J.: Turán problems and shadows I: paths and cycles. J. Comb. Theory Ser. A 129, 57–79 (2015)

    Article  MATH  Google Scholar 

  13. Lan, Y., Shi, Y., Song, Z-X.: Extremal Theta-free planar graphs. arXiv:1711.01614

  14. Lan, Y., Shi, Y., Song, Z.-X.: Extremal \(H\)-free planar graphs. Electron. J. Comb. 26, 2.11 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Turán, P.: Eine Extremalaufgabe aus der Graphentheorie, (Hungarian) Mat. Fiz. Lapok 48, 436–452 (1941)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for many valuable suggestions and comments, especially detailed proof idea for Theorems 1.3, 1.4 and 1.5, which also prompts the addition of a new result (Theorem 1.6). The authors also would like to thank Zi-Xia Song for helpful discussion and Shunyu Yao for his help on the computer program. This work was partially supported by National Natural Science Foundation of China, Natural Science Foundation of Tianjin (no. 17JCQNJC00300), the China-Slovenia bilateral project “Some topics in modern graph theory” (no. 12-6), Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (no. CICIP2018005), and the Fundamental Research Funds for the Central Universities, Nankai University (63191516). Funding was provided by National Natural Science Foundation of China (11771221, 11811540390).

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin, 300387, China

    Yongxin Lan

  2. Center for Combinatorics and LPMC, Nankai University, Tianjin, 300071, China

    Yongtang Shi

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  1. Yongxin Lan
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  2. Yongtang Shi
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Correspondence to Yongtang Shi.

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Appendices

Appendix 1

List of 18 graphs satisfying the three properties in claim used in the proof of Theorem 1.5.

figure a

Appendix 2

List of 200 graphs satisfying the three properties in claim used in the proof of Theorem 1.6.

figure b
figure c
figure d
figure e
figure f

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Lan, Y., Shi, Y. Planar Turán Numbers of Short Paths. Graphs and Combinatorics 35, 1035–1049 (2019). https://doi.org/10.1007/s00373-019-02055-w

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  • DOI: https://doi.org/10.1007/s00373-019-02055-w

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