Planar Tur\'an number of disjoint union of $C_3$ and $C_5$

TL;DR

This paper determines the maximum number of edges in large planar graphs that avoid containing a disjoint union of a triangle and a pentagon, providing an exact formula and characterizing extremal graphs.

Contribution

It establishes the exact planar Turán number for the union of a triangle and a pentagon and characterizes extremal graphs for large n.

Findings

Exact formula for ex_{\

Characterization of extremal graphs for large n

Extension of known results to new cycle unions

Abstract

The planar Tur\'an number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Tur\'an number of $k\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\ell}$ denote the cycle of length $\ell$ and $C_{\ell}\cup C_t$ denote the union of disjoint cycles $C_{\ell}$ and $C_t$. The planar Tur\'an number $ex_{\mathcal{P}}(n,H)$ is known if $H=C_{\ell}\cup C_k$, where $\ell,k\in \{3,4\}$. In this paper, we determine the value $ex_{\mathcal{P}}(n,C_3\cup C_5)=\lfloor\frac{8n-13}{3}\rfloor$ and characterize the extremal graphs when $n$ is sufficiently large.