Planar Turán Number of the 6-Cycle

Let exP(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=K₂, exP(n,T,H) is the well-studied function, the planar Turán number of H, denoted by exP(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 exP(n,C₄) and exP(n,C₅). Later on, Lan, Shi, and Song continued this topic and proved that exP(n,C₆)≤ 18(n-2)/7. In this paper, we give a sharp upper bound exP(n,C₆) ≤ 5/2n-7, for all n≥ 18, which improves Lan, Shi, and Song's result. We also pose a conjecture on exP(n,Cₖ), for k≥ 7.