Extremal planar graphs with no cycles of particular lengths

Abstract

In this paper we estimate the planar Turán number ${\mathrm{ex}}_{\mathcal{P}}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.