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DOI: 10.7155/jgaa.00516
On the Circumference of Essentially 4-connected Planar Graphs
Igor Fabrici, Jochen Harant, Samuel Mohr, and Jens M. Schmidt
Vol. 24, no. 1, pp. 21-46, 2020. Regular paper.
Abstract A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.
Submitted: February 2019.
Reviewed: August 2019.
Revised: September 2019.
Accepted: December 2019.
Final: December 2019.
Published: January 2020.
Communicated by Giuseppe Liotta
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