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 .