Planar Tur\'an number of two adjacent cycles

TL;DR

This paper determines the maximum number of edges in large planar graphs that avoid containing two specific adjacent cycles, specifically for the cases of two triangles and a triangle and a quadrilateral connected by an edge.

Contribution

It provides exact values for the planar Turán number of graphs formed by two adjacent cycles, specifically for C3-C3 and C3-C4 configurations.

Findings

Determined the planar Turán number for C3-C3.

Determined the planar Turán number for C3-C4.

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 $k(k\geq 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Tur\'an number of $C_{3}\text{-}C_{3}$ and $C_{3}\text{-}C_{4}$, where $C_{k}\text{-}C_{\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them.