Faces in girth-saturated graphs on surfaces

TL;DR

This paper investigates the maximum length of facial cycles in girth-saturated graphs embedded on surfaces, establishing bounds and asymptotic behavior related to surface genus and girth.

Contribution

It provides new bounds for facial cycle lengths in girth-saturated graphs on surfaces, including exact bounds for the plane and asymptotic relations for general surfaces.

Findings

Bounded ${ m f}_{ m max}( ext{ell}, ext{surface})$ for all surfaces and girths.

${ m f}_{ m max}( ext{ell}, ext{surface})$ grows linearly with genus for fixed girth.

For the plane, ${ m f}_{ m max}( ext{ell}, ext{plane})$ is between $3 ext{ell}-11$ and $8 ext{ell}-13$.

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 $\Sigma$.