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Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an 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 asubgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write$ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs wasinitiated by Dowden (2016). He obtained sharp upper bound for both$ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtainedsharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upperbounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and$ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructionswhich show the bounds are tight for infinitely many graphs.

The planar Tur\'an number of $\{K_4,C_5\}$ and $\{K_4,C_6\}$