Faces in girth-saturated graphs on surfaces

Abstract

What is the maximum length ${\rm f}_{\rm max}(\ell, Σ)$ of a facial cycle of an inclusion-maximal graph with girth at least $\ell$ embedded on a given surface $Σ$? If $Σ=\mathcal{P}$ 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 ${\rm f}_{\rm max}(\ell, Σ)$ is bounded for any integer $\ell$ and any closed surface $Σ$. For a fixed $Σ$, we show that $Ω(\ell) ={\rm f}_{\rm max}(\ell, Σ) = O(\ell^2)$, while for a fixed $\ell\ge 6$, ${\rm f}_{\rm max}(\ell, Σ)=Θ(g)$, where $g$ is the genus of $Σ$.

Figures (11)

• Figure 1: An illustration to the definitions of $C$-convex path and lens.
• Figure 2: An illustration to the proof of \ref{['conv_geod']}.
• Figure 3: The set $\{x',y'\}$ can separate $x$ and $y$ in $C$ (left) or not (right).
• Figure 4: An illustration to Case 1.
• Figure 5: An illustration to Case 2.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof