Longer Cycles in Essentially 4Connected Planar Graphs
Abstract
A planar 3connected graph $G$ is called \emph{essentially $4$connected} if, for every 3separator $S$, at least one of the two components of $GS$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm circ}\nolimits(G)$ of a longest cycle of any essentially 4connected 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 4connected 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 $O(n^2)$.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.05619
 arXiv:
 arXiv:1710.05619
 Bibcode:
 2017arXiv171005619F
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05C38;
 05C10
 EPrint:
 Discussiones Mathematicae Graph Theory, ISSN (Online) 20835892