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arXiv · 1806.09413

On the Circumference of Essentially 4-connected Planar Graphs

Igor Fabrici·Jochen Harant·Samuel Mohr·Jens M. Schmidt
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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 $\frac{3}{5}(n+2)$.

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Igor Fabrici, Jochen Harant, Samuel Mohr, Jens M. Schmidt. 2018-06-25. On the Circumference of Essentially 4-connected Planar Graphs. https://arxiv.org/abs/1806.09413

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