Induced planar Tur\'an numbers

TL;DR

This paper introduces the concept of induced planar Turán numbers, establishing bounds and exact values for specific forbidden induced subgraphs in planar graphs.

Contribution

It defines induced planar Turán numbers and determines sharp bounds and exact extremal values for certain small forbidden induced subgraphs.

Findings

Sharp upper bound for $F=\Theta_4$

Exact extremal values for paths $P_3$, $P_4$, and $P_5$

Introduces a new variation of Turán numbers in planar graphs

Abstract

The planar Tur\'a number of a graph $F$ is the maximum number of edges an $n$-vertex $F$-free planar graph can have. We study the case where $F$ is forbidden as an induced subgraph, thereby introducing the induced planar Tur\'a numbers. We will determine a sharp upper bound when $F$ is $\Theta_4$, a $4$-cycle with a diagonal edge, and obtain exact extremal values in case $F$ is a path $P_k$ on $k$ vertices, for $k=3,4$ and $5$.