Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 2
Planar Turán Number of the Θ6
Authors:
Chuanqi Xiao School of Science, Nanjing University of Posts and Telecommunications, Nanjing, P. R. China

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,
Debarun Ghosh Central European University, Budapest, Hungary

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,
Ervin Győri Central European University, Budapest, Hungary
Hun-Ren Alfréd Rényi Institute of Mathematics, Budapest, Hungary

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,
Addisu Paulos Central European University, Budapest, Hungary
Addis Ababa University, Addis Ababa, Ethiopia

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, and
Oscar Zamora Universidad de Costa Rica, San José, Costa Rica

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Pages:
89–115
Online Publication Date:
24 Jul 2024
Publication Date:
24 Jul 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2024.04307
Keywords:
Planar Turán number; 𝚯𝒃 extremal planar graph
Restricted access

Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.

Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.

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    C. Dowden. Extremal 𝐶4-free/𝐶5-free planar graphs. Journal of Graph Theory, 83:213–230, 2016.

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    P. Erdős. On the structure of linear graphs. Israel Journal of Mathematics, 1:156–160, 1963.

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    P. Erdős. On the number of complete subgraphs contained in certain graphs. Publ. Math. Inst. Hung. Acad. Sci., 7:459–464, 1962.

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    L. Fang, B. Wang, and M. Zhai. Planar Turán number of intersecting triangles. Discrete Math., 345(5):112794, 2022.

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    D. Ghosh, E. Győri, A. Paulos, and C. Xiao. Planar Turán Number of Double Stars. https://arxiv.org/abs/2006.00994, 2022

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    D. Ghosh, E. Győri, R. R. Martin, A. Paulos, and C. Xiao. Planar Turán Number of the 6-Cycle. SIAM Journal on Discrete Mathematics, 36(3):2028–2050, 2022.

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    Y. Lan, Y. Shi, and Z.-X. Song. Extremal 𝐻-free planar graphs. The Electronic Journal of Combinatorics, 26(2):#P2.11, 2019.

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    Y. Lan, Y. Shi, and Z.-X. Song. Extremal theta-free planar graphs. Discrete Mathematics, 342(12):no. 111610, 2019.

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    A. Zykov. On some properties of linear complexes. Mat. Sb. (N.S.), 24(66):163–188, 1949.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

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Gábor SÁGI (Rényi Institute of Mathematics)

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  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
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  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
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  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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