Planar Turán Numbers of Cubic Graphs and Disjoint Union of Cycles

• Published in: Graphs and combinatorics Vol. 40; no. 2
• Main Authors: Lan, Yongxin, Shi, Yongtang, Song, Zi-Xia
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.04.2024; Springer Nature B.V
• Subjects: Apexes; Combinatorics; Engineering Design; Graph theory; Lower bounds; Mathematics; Mathematics and Statistics; Original Paper; Turán number; 05C10; Planar triangulation; 05C35; Extremal planar graph
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-024-02750-3

The planar Turán number of a graph H , denoted by e x P ( n , H ) , is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding e x P ( n , H ) when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of e x P ( n , H ) when H is a cubic graph. We then prove that e x P ( n , 2 C 3 ) = ⌈ 5 n / 2 ⌉ - 5 for all n ≥ 6 , and obtain the lower bounds of e x P ( n , 2 C k ) for all n ≥ 2 k ≥ 8 . Finally, we also completely determine the exact values of e x P ( n , K 2 , t ) for all t ≥ 3 and n ≥ t + 2 .