Dual-Diameter Bounds for Triangulated Surfaces

Definitions and mathematical statement

For a finite triangulation $T$ of a surface, define its dual graph $G_T$ to have one vertex for each triangle of $T$, with two distinct vertices adjacent when their triangles share an edge. If $G_T$ is connected, its dual diameter is \[ \operatorname{diam}(G_T)=\max_{\sigma,\tau\in V(G_T)}d_{G_T}(\sigma,\tau), \] where $d_{G_T}$ is shortest-path distance.

Let $\mathcal S_n$ denote the intended class of triangulated surfaces with boundary having exactly $n$ vertices. The supplied record describes $\overline f(2,n)$ as the maximal dual diameter in this class. Thus, whenever the class and its maximum are well defined, \[ \overline f(2,n)=\max_{T\in\mathcal S_n}\operatorname{diam}(G_T). \] The proposed question is whether, for every admissible integer $n$, \[ \overline f(2,n)\leq n-3. \] Equivalently, once the intended class is specified, the proposed bound is $\operatorname{diam}(G_T)\leq |V(T)|-3$ for every $T$ in that class.

Unresolved qualifications. The record does not specify the precise class $\mathcal S_n$, including the triangulation convention, connectedness, allowed topology, whether the boundary must be nonempty, or whether interior vertices are allowed. It also does not specify the range of $n$. The dual-graph definition above supplies a conventional interpretation; its agreement with the cited source requires review. Consequently, this is a conditional formulation of the supplied question, pending clarification of its domain.

Short English statement

Does every triangulated surface in the intended class, with $n$ vertices, have dual diameter at most $n-3$? The allowed surfaces and triangulations require clarification. The supplied record labels the question open, but its dated literature assessment discusses a different topic and does not establish the status of this bound.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002067$, title Dual-Diameter Bounds for Triangulated Surfaces, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year $2012$ and supplies no named proposer.
• Supplied source citation: Triangulations (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/24, pp. 1405--1486. Supplied source link: https://doi.org/10.4171/owr/2012/24. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This citation has not been independently verified here.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2012/25. This DOI differs from the source citation above; the record does not resolve the discrepancy.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. Its supplied status is open. The assessment describes an incomplete work-in-progress paragraph about Lagrangian structures and the Novikov conjecture, states that it is not a complete claim whose truth can be checked, and retains an open label under its stated rule. That description does not match the dual-diameter question. The background explicitly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is self-contained and adds no unsupported source-dependent content. These are supplied review claims, not independent mathematical or source verification by this draft.