Planar Tur\'{a}n Number of intersecting triangles

TL;DR

This paper determines the maximum number of edges in large planar graphs that avoid certain intersecting triangle configurations, specifically friendship graphs and related structures, improving previous bounds in the literature.

Contribution

The paper provides sharp bounds for the planar Turán number of friendship graphs and related graphs, advancing the understanding of extremal planar graph structures.

Findings

Established sharp bounds for $ex_{ ext{P}}(n,H_k)$ and $ex_{ ext{P}}(n,K_1+P_{k+1})$.

Improved previous results by Lan and Shi on planar Turán numbers.

Enhanced understanding of extremal structures avoiding intersecting triangles.

Abstract

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph, which is obtained from $k$ triangles by sharing a common vertex. In this paper, we obtain sharp bounds of $ex_{\mathcal{P}}(n,H_k)$ and $ex_{\mathcal{P}}(n,K_1+P_{k+1})$ for $k\ge2$, which improves the results of Lan and Shi in Electron. J. Combin. 26 (2) (2019), \#P2.11.