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Structure and pancyclicity of maximal planar graphs with diameter two

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Abstract

A graph G on n vertices is called non-universal if its maximum degree is at most \(n-2\). In this paper, we give a structural characterization for non-universal maximal planar graphs with diameter two. In precise, we find 10 basic graphs, and then generate all 25 non-universal maximal planar graphs with diameter two by adding repeatedly and appropriately 3-vertices to some of these 10 basic graphs. As an application, we show that maximal planar graphs with diameter two are pancyclic except five special graphs.

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References

  • Hakimi SL, Schmeichel EF (1979) On the number of cycles of length \(k\) in a maximal planar graph. J Graph Theory 3:69–86

    Article  MathSciNet  Google Scholar 

  • Hell P, Seyffarth K (1993) Largest planar graphs of diameter two and fixed maximum degree. Discrete Math 111:313–322

    Article  MathSciNet  Google Scholar 

  • Holton DA, McKay BD (1988) The smallest non-Hamiltonian 3-connected cubic planar graphs have 38 vertices. J Combin Theory Ser B 45:305–319

    Article  MathSciNet  Google Scholar 

  • Huang D, Wang Y, Lv J, Yang Y, Wang W (2019) List coloring and diagonal coloring for plane graphs of diameter two. Appl Math Comput 363: Article 124562, 10 pp

  • MacGillivray G, Seyffarth K (1996) Domination numbers of planar graphs. J Graph Theory 22:213–229

    Article  MathSciNet  Google Scholar 

  • Seyffarth K (1989) Maximal planar graphs of diameter two. J Graph Theory 13:619–648

  • Tutte WT (1956) A theorem on planar graphs. Trans Am Math Soc 82:99–116

    Article  MathSciNet  Google Scholar 

  • Wang Y, Cui SY, Wang W (2020) Wegner’s conjecture is true for maximal planar graphs with diameter two, Submitted

  • Wang Y, Zheng L (2020) Computation on the difference of Zagreb indices of maximal planar graphs with diameter two. Appl Math Comput 377: Artical 125187, 13 pp

  • Whitney H (1931) A theorem on graphs. Ann Math 32:378–390

    Article  MathSciNet  Google Scholar 

  • Yang A, Yuan J (2007) On the vertex arboricity of planar graphs of diameter two. Discrete Math 307:2438–2447

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Shu-Yu Cui, Danjun Huang & Weifan Wang

  2. School of Management, Beijing University of Chinese Medicine, Beijing, 100029, China

    Yiqiao Wang

  3. Department of Computer Science and Technology, Harbin Institute of Technology, Shenzhen, China

    Hongwei Du

Authors
  1. Shu-Yu Cui
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  2. Yiqiao Wang
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  3. Danjun Huang
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  4. Hongwei Du
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  5. Weifan Wang
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Corresponding author

Correspondence to Weifan Wang.

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S.-Y. Cui: Research supported by NSFC (No. 11801521).

Y. Wang: Research supported by NSFC (No. 12071048) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

D. Huang: Research supported by ZJNSF (No. LY18A010014).

W. Wang: Research supported by NSFC (Nos. 11771402; 12031018)

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Cui, SY., Wang, Y., Huang, D. et al. Structure and pancyclicity of maximal planar graphs with diameter two. J Comb Optim 43, 1–27 (2022). https://doi.org/10.1007/s10878-021-00749-7

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  • DOI: https://doi.org/10.1007/s10878-021-00749-7

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