Faces in girth-saturated graphs on surfaces

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 $\Sigma$.