The planar Turán number of $\{K_4,{\Theta }_6^i\}$

Abstract

Let $\mathcal{H}$ be a family of graphs. A graph is said to be $\mathcal{H}$-free if it contains no subgraph isomorphic to a graph in $\mathcal{H}$. The planar Turán number $e{x}_{{}_{\mathcal{P}}}(n,\mathcal{H})$ is defined as the maximum number of edges in an $\mathcal{H}$-free planar graph on $n$ vertices. In this paper, we determine the exact value of $e{x}_{{}_{\mathcal{P}}}(n,\{K_4,{\Theta }_6^1\})$ and a tight upper bound of $e{x}_{{}_{\mathcal{P}}}(n,\{K_4,{\Theta }_6^2\})$.