Longer Cycles in Essentially 4-Connected Planar Graphs

TL;DR

This paper proves that essentially 4-connected planar graphs on n vertices contain longer cycles of at least 3/5 of n, improving previous bounds, and provides an efficient algorithm to find such cycles.

Contribution

It establishes a new lower bound of 3/5(n+2) for the longest cycle in essentially 4-connected planar graphs and offers an O(n^2) algorithm to find such cycles.

Findings

Longest cycle length at least 3/5 of n+2

Improved bound over previous results

Cycle can be found in quadratic time

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 $\mathop{\rm circ}\nolimits(G)$ 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 $O(n^2)$.