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arXiv · 2410.13481

Faces in girth-saturated graphs on surfaces

Maria Axenovich·Leon Kießle·Arsenii Sagdeev·Maksim Zhukovskii
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Abstract

What is the maximum length ${\rm f}_{\rm max}(\ell, \Sigma)$ of a facial cycle of an inclusion-maximal graph with girth at least $\ell$ embedded on a given surface $\Sigma$? If $\Sigma=\mathcal{P}$ is a plane, we show that $3\ell-11\leq {\rm f}_{\rm max}(\ell, \mathcal{P})\leq 8\ell-13$. We also prove that ${\rm f}_{\rm max}(\ell, \Sigma)$ is bounded for any integer $\ell$ and any closed surface $\Sigma$. For a fixed $\Sigma$, we show that $\Omega(\ell) ={\rm f}_{\rm max}(\ell, \Sigma) = O(\ell^2)$, while for a fixed $\ell\ge 6$, ${\rm f}_{\rm max}(\ell, \Sigma)=\Theta(g)$, where $g$ is the genus of $\Sigma$.

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Maria Axenovich, Leon Kießle, Arsenii Sagdeev, Maksim Zhukovskii. 2024-10-17. Faces in girth-saturated graphs on surfaces. https://arxiv.org/abs/2410.13481

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