Planar Turán Numbers of Cubic Graphs and Disjoint Union of Cycles

Abstract

The planar Turán number of a graph H, denoted by \(ex_{_\mathcal {P}}(n,H)\) , 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 \(ex_{_\mathcal {P}}(n,H)\) 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 \(ex_{_\mathcal {P}}(n,H)\) when H is a cubic graph. We then prove that \(ex_{_\mathcal {P}}(n,2C_3)=\lceil 5n/2\rceil -5\) for all \(n\ge 6\) , and obtain the lower bounds of \(ex_{_\mathcal {P}}(n,2C_k)\) for all \(n\ge 2k\ge 8\) . Finally, we also completely determine the exact values of \(ex_{_\mathcal {P}}(n,K_{2,t})\) for all \(t\ge 3\) and \(n\ge t+2\) .