The planar Tur\'an number of $\{K_4,C_5\}$ and $\{K_4,C_6\}$

TL;DR

This paper establishes upper bounds on the maximum edges in planar graphs avoiding certain subgraphs, specifically for sets involving $K_4$ combined with $C_5$ and $C_6$, and demonstrates these bounds are tight for infinitely many cases.

Contribution

It provides new tight upper bounds for the planar Turán numbers of $ig ext{set}igrace{ ext{K}_4, ext{C}_5}$ and $ig ext{set}igrace{ ext{K}_4, ext{C}_6}$, extending previous extremal graph results.

Findings

Upper bound for $ex_ ext{P}(n, ext{K}_4, ext{C}_5)$ is $rac{15}{7}(n-2)$.

Upper bound for $ex_ ext{P}(n, ext{K}_4, ext{C}_6)$ is $rac{7}{3}(n-2)$.

Bounds are tight for infinitely many graphs.

Abstract

Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.