The Planar Turán Number of \(\{K_4,C_5\}\) and \(\{K_4,C_6\}\)

Let \(\mathcal {H}\) $\mathcal{H}$ be a set of graphs. The planar Turán number, \(ex_\mathcal {P}(n,\mathcal {H})\) $e{x}_{\mathcal{P}}(n,\mathcal{H})$ , is the maximum number of edges in an n-vertex planar graph which does not contain any member of \(\mathcal {H}\) $\mathcal{H}$ as a subgraph. When \(\mathcal {H}=\{H\}\) $\mathcal{H}=\{H\}$ has only one element, we usually write \(ex_\mathcal {P}(n,H)\) $e{x}_{\mathcal{P}}(n,H)$ instead. The study of extremal planar graphs was initiated by Dowden (J Graph Theory 83(3):213–230, 2016). He obtained sharp upper bounds for both \(ex_\mathcal {P}(n,C_5)\) $e{x}_{\mathcal{P}}(n,C_5)$ and \(ex_\mathcal {P}(n,K_4)\) $e{x}_{\mathcal{P}}(n,K_4)$ . Later on, sharp upper bounds were proved for \(ex_\mathcal {P}(n,C_6)\) $e{x}_{\mathcal{P}}(n,C_6)$ and \(ex_\mathcal {P}(n,C_7)\) $e{x}_{\mathcal{P}}(n,C_7)$ . In this paper, we show that \(ex_\mathcal {P}(n,\{K_4,C_5\})\le {15\over 7}(n-2)\) $e{x}_{\mathcal{P}}(n,\{K_4,C_5\})\le \frac{15}{7}(n-2)$ and \(ex_\mathcal {P}(n,\{K_4,C_6\})\le {7\over 3}(n-2)\) $e{x}_{\mathcal{P}}(n,\{K_4,C_6\})\le \frac{7}{3}(n-2)$ . We also give constructions which show the bounds are sharp for infinitely many n.