Extremal planar graphs with no cycles of particular lengths

TL;DR

This paper investigates the maximum number of edges in planar graphs that avoid certain subgraphs, providing new proofs and extending the 'contribution method' to estimate planar Turán numbers for specific cycles.

Contribution

It offers a new, concise proof for the case when H is a 5-cycle and extends the 'contribution method' to bipartite and triangle-free graphs avoiding short even cycles.

Findings

New proof for planar Turán number when H=C5

Extension of the 'contribution method' to bipartite and triangle-free graphs

Estimates for maximum edges in graphs avoiding specific cycles

Abstract

In this paper we estimate the planar Tur\'an 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, Gy\H{o}ri, Martin, Paulos and Xiao in arXiv:2004.14094.