The planar Turán number of $\{K_{4},Θ_{6}^{i}\}$

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 $ex_{_\mathcal{P}}(n,\mathcal{H})$ 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 $ex_{_\mathcal{P}}(n,\{K_{4}, Θ_{6}^{1}\})$ and a tight upper bound of $ex_{_\mathcal{P}}(n,\{K_{4}, Θ_{6}^{2}\})$.