Faces in Girth-Saturated Graphs on Surfaces

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