Planar turan number of the 6-cycle

Let ex(P)(n, T, H) denote the maximum number of copies of T in an n-vertex planar graph which does not contain H as a subgraph. When T = K-2, ex(P)(n, T, H) is the well-studied function, the planar Turan number of H, denoted by ex(P)(n, H). The topic of extremal planar graphs was initiated by Dowden [J. Graph Theory, 83 (2016), pp. 213-230]. He obtained a sharp upper bound for both ex(P)(n, C-4) and ex(P)(n, C-5). Later on, Lan, Shi, and Song continued this topic and proved that ex(P)(n, C-6) <= 18(n-2)/7. In this paper, we give a sharp upper bound ex(P)(n, C-6) <= 5/2n - 7, for all n >= 18, which improves Lan, Shi, and Song's result. We also pose a conjecture on ex(P)(n, C-k), for k >= 7.