Longer Cycles in Essentially 4-Connected Planar Graphs

Abstract

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 $\mathrm{circ}(G)$ 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 $\mathrm{circ}(G)\ge \frac{1}{2}(n+4)$. 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 $O(n^2)$.

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• Large cycles in essentially 4-connected graphs