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

TL;DR

This paper investigates the maximum number of edges in planar graphs that avoid certain cubic graphs, disjoint cycles, or complete bipartite graphs, extending the understanding of planar Turán numbers for these specific subgraphs.

Contribution

It provides new bounds and results for the planar Turán number when the forbidden subgraph is a cubic graph, a disjoint union of cycles, or a complete bipartite graph.

Findings

Derived bounds for $ex_{\mathcal{P}}(n,H)$ with $H$ as cubic graphs.

Analyzed $ex_{\mathcal{P}}(n,H)$ for disjoint unions of cycles.

Extended known results to new classes of forbidden subgraphs.

Abstract

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 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 study $ex_{_\mathcal{P}}(n,H)$ when $H$ is a cubic graph or disjoint union of cycles or $H=K_{s, t}$.