Asymptotically sharp oriented-diameter bound for planar triangulations

Establish whether there exists an absolute constant C such that every n-vertex planar triangulation G satisfies ordiam(G) <= n/3 + C, thereby matching the asymptotic lower bound furnished by the nested-triangle triangulations.

Background

The paper studies the oriented diameter of planar triangulations, defined as the minimum diameter over all strong orientations of a connected bridgeless graph. Before this work, nested-triangle triangulations supplied examples with oriented diameter at least n/3, while the best general upper bound was approximately n/2.

The authors improve the general upper bound to (2n+44)/5, but explicitly do not attain the conjectured n/3+O(1) bound. They prove the conjecture up to an additive constant for 4-connected planar triangulations, leaving the unrestricted case unresolved.

References

These results naturally suggest the following conjecture. There exists an absolute constant C such that every n-vertex planar triangulation G satisfies \ordiam(G)\leq \frac n3+C. Conjecture~\ref{conj:main}, if true, would be asymptotically best possible. Although we do not resolve the conjecture in this paper, we improve the general upper bound from n/2+O(1) to 2n/5+O(1).

— Improved bounds on the oriented diameter of planar triangulations  (2608.19089 - Liu, 19 Aug 2026) in Conjecture 1 (labelled Conjecture~\ref{conj:main}), Section 1, Introduction