Faces in Girth-Saturated Graphs on Surfaces

What is the maximum length fₘax( , Σ ) f max ( ℓ , Σ ) of a facial cycle of an inclusion-maximal graph with girth at least ℓ embedded on a given surface Σ Σ ? If Σ =P Σ = P is a plane, we show that 3 -11≤ fₘax( , P)≤ 8 -13 3 ℓ - 11 ≤ f max ( ℓ , P ) ≤ 8 ℓ - 13 . We also prove that fₘax( , Σ ) f max ( ℓ , Σ ) is bounded for any integer ℓ and any closed surface Σ Σ . For a fixed Σ Σ , we show that Ω ( ) =fₘax( , Σ ) = O( ²) Ω ( ℓ ) = f max ( ℓ , Σ ) = O ( ℓ 2 ) , while for a fixed ≥ 6 ℓ ≥ 6 , fₘax( , Σ )=Θ (g) f max ( ℓ , Σ ) = Θ ( g</mml