Circumference of essentially 4-connected planar triangulations - Fabrici, Igor; Harant, Jochen; Mohr, Samuel; Schmidt, Jens (Journal of Graph Algorithms and Applications, 2021)
A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains atleast two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least$\frac{2}{3}(n+4)$; moreover, this bound is sharp.