A uniform linear bound for facial cycles on surfaces

Let $G$ be a finite simple graph of girth at least $\ell\ge3$, maximal in its fixed crossing-free embedding on an orientable surface with $g\ge1$ handles or a nonorientable surface with $g\ge1$ crosscaps. We prove that every simple cycle forming the entire boundary of a face has length at most $100000g\ell$. The same constant works for both surface types, including noncellular embeddings. The proof bounds the patterns of equal-radius balls along one facial cycle by counting alternating disjoint trees. Published hypergraph covering estimates then provide few centers, whose boundary runs are short by the girth condition.