Planar Tur\'an number of quasi-double stars

TL;DR

This paper investigates the maximum number of edges in planar graphs that avoid certain quasi-double star subgraphs, providing tight bounds and exact values for various parameters.

Contribution

It establishes new bounds and exact values for the planar Turán number of (h,k)-quasi-double stars, expanding understanding of extremal planar graphs.

Findings

Derived tight bounds for ex_{ ext{P}}(n, W_{h,k}) when 3 ≤ h+k ≤ 5.

Established exact extremal values for specific (h,k) cases such as (1,5), (2,4), and (2,5).

Provided asymptotic bounds for the planar Turán number of certain quasi-double star graphs.

Abstract

Given a graph H, we call a graph $\textit{H-free}$ if it does not contain H as a subgraph. The planar Tur\'an number of a graph H, denoted by $ex_{\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1v_2v_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $ex_{\mathcal{P}}(n,W_{h,k})$ for all $1\le h\le 2\le k\le 5$, and obtain some tight bounds $ex_{\mathcal{P}}(n,W_{h,k})\leq\frac{3(h+k)}{h+k+2}n$ for $3\le h+k\le 5$ with equality holds if $(h+k+2)\mid n$, and $ex_{\mathcal{P}}(n,W_{1,5})\le \frac{5}{2}n$ with equality holds if $12\mid n$. Also we show that $\frac{9}{4}n\le ex_{\mathcal{P}}(n,W_{2,4})\le \frac{5}{2}n$ and $\frac{5}{2}n\le ex_{\mathcal{P}}(n,W_{2,5})\le \frac{17}{6}n$, respectively.