Planar Turán Numbers of Cubic Graphs and Disjoint Union of Cycles.

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 . [ABSTRACT FROM AUTHOR]