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The planar Tur\'an number of a graph $H$, denoted $ex_{_\mathcal{P}}(n,H)$,is the maximum number of edges in a planar graph on $n$ vertices withoutcontaining $H$ as a subgraph. This notion was introduced by Dowden in 2016 andhas attracted quite some attention since then; those work mainly focus onfinding $ex_{_\mathcal{P}}(n,H)$ when $H$ is a cycle or Theta graph or $H$ hasmaximum degree at least four. In this paper, we study $ex_{_\mathcal{P}}(n,H)$when $H$ is a cubic graph or disjoint union of cycles or $H=K_{s, t}$.

Planar Tur\'an numbers of cubic graphs and disjoint union of cycles