Geodesic
Circumference of essentially 4-connected planar triangulations
Journal of Graph Algorithms and Applications, Tome 25 (2021) no. 1, pp. 121-132.

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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 at least 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.
DOI : 10.7155/jgaa.00552
Keywords: circumference, long cycle, triangulation, essentially 4-connected, planar graph 
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Igor Fabrici; Jochen Harant; Samuel Mohr; Jens Schmidt. Circumference of essentially 4-connected planar triangulations. Journal of Graph Algorithms and Applications, Tome 25 (2021) no. 1, pp. 121-132. doi : 10.7155/jgaa.00552. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00552/

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