Planar Tur\'{a}n number of disjoint union of $C_3$ and $C_4$

TL;DR

This paper determines the exact maximum number of edges in planar graphs that avoid a disjoint union of a triangle and a quadrilateral, extending previous results and characterizing extremal graphs.

Contribution

It provides the exact planar Turán number for disjoint unions of $C_3$ and $C_4$, and improves bounds for larger cycles.

Findings

Exact value of $ex_{\\mathcal{P}}(n, C_3 \cup C_4)$ determined.

Extremal graphs characterized for the union of $C_3$ and $C_4$.

Lower bounds improved for large cycle unions.

Abstract

The {\em planar Tur\'{a}n number} of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $H$-free planar graph. The planar Tur\'{a}n number of $k\geq 3$ vertex-disjoint union of cycles is a trivial value $3n-6$. Lan, Shi and Song determine the exact value of $ex_{\mathcal{P}}(n,2C_3)$. We continue to study planar Tur\'{a}n number of vertex-disjoint union of cycles and obtain the exact value of $ex_{\mathcal{P}}(n,H)$, where $H$ is vertex-disjoint union of $C_3$ and $C_4$. The extremal graphs are also characterized. We also improve the lower bound of $ex_{\mathcal{P}}(n,2C_k)$ when $k$ is sufficiently large.