Planar Turán numbers of cubic graphs and disjoint union of cycles

Abstract

The planar Turán number of a graph $H$, denoted $e{x}_{{}_{\mathcal{P}}}(n,H)$, 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 $e{x}_{{}_{\mathcal{P}}}(n,H)$ when $H$ is a cycle or Theta graph or $H$ has maximum degree at least four. In this paper, we study $e{x}_{{}_{\mathcal{P}}}(n,H)$ when $H$ is a cubic graph or disjoint union of cycles or $H=K_{s,t}$.

The second version adds an improved lower bound for the planar Turan number of disjoint union of cycles of length at least seven, see Lemma 5.2. arXiv admin note: text overlap with arXiv:1808.01487