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

Let \(\mathcal {H}\) be a set of graphs. The planar Turán number, \(ex_\mathcal {P}(n,\mathcal {H})\), is the maximum number of edges in an n-vertex planar graph which does not contain any member of \(\mathcal {H}\) as a subgraph. When \(\mathcal {H}=\{H\}\) has only one element, we usually write \(ex_\mathcal {P}(n,H)\) 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 \(ex_\mathcal {P}(n,C_5)\) and \(ex_\mathcal {P}(n,K_4)\). Later on, sharp upper bounds were proved for \(ex_\mathcal {P}(n,C_6)\) and \(ex_\mathcal {P}(n,C_7)\). In this paper, we show that \(ex_\mathcal {P}(n,\{K_4,C_5\})\le {15\over 7}(n-2)\) and \(ex_\mathcal {P}(n,\{K_4,C_6\})\le {7\over 3}(n-2)\). We also give constructions which show the bounds are sharp for infinitely many n.