Planar Turán number of two adjacent cycles

Abstract

The planar Turán number of $H$, denoted by $e{x}_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k(k\ge 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Turán 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.

18 pages, 7 figures