  A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if,
for every 3-separator $S$, at least one of the two components of $G-S$ is an
isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm
circ}\nolimits(G)$ of a longest cycle of any essentially 4-connected planar
graph $G$ on $n$ vertices is at least $\frac{2n+4}{5}$ and Fabrici, Harant and
Jendrol' improved this result to $\mathop{\rm circ}\nolimits(G)\geq
\frac{1}{2}(n+4)$. In the present paper, we prove that an essentially
4-connected planar graph on $n$ vertices contains a cycle of length at least
$\frac{3}{5}(n+2)$ and that such a cycle can be found in time $O(n^2)$.

Combinatorics;Discrete Mathematics