Large cycles in essentially 4-connected graphs

TL;DR

This paper improves the lower bound on the length of the longest cycle in essentially 4-connected planar graphs to a tight bound of rac{2n+6}{3}or all n6, advancing understanding of cycle structures in such graphs.

Contribution

The paper establishes a new tight lower bound for the longest cycle in essentially 4-connected planar graphs, refining previous bounds and extending Thomassen's results.

Findings

Improved lower bound to rac{2n+6}{3}or cycle length

Bound is proven to be best possible

Extends Thomassen's results on Tutte paths

Abstract

Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but there are 3-connected $n$-vertex planar graphs whose longest cycles have length $\Theta(n^{\log_32})$. On the other hand, Jackson and Wormald in 1992 proved that an essentially 4-connected $n$-vertex planar graph contains a cycle of length at least $(2n+4)/5$, which was recently improved to $5(n+2)/8$ by Fabrici {\it et al}. In this paper, we improve this bound to $\lceil (2n+6)/3\rceil$ for $n\ge 6$, which is best possible, by proving a quantitative version of a result of Thomassen on Tutte paths.