Improved bounds on the oriented diameter of planar triangulations

Abstract: The oriented diameter of a connected bridgeless graph $G$, denoted by $\overrightarrow{\operatorname{diam}}(G)$, is the minimum diameter among all strong orientations of $G$. We study the oriented diameter of planar triangulations, and show that $$\overrightarrow{\operatorname{diam}}(G)\leq \frac{2n+44}{5}$$ for any $n$-vertex planar triangulation $G$. This improves the leading constant in the previous best general upper bound $\lceil \frac{n}{2}\rceil$, due to Ge, Liu, and Wang, from $1/2$ to $2/5$. We also prove that every $n$-vertex $4$-connected planar triangulation satisfies $$\overrightarrow{\operatorname{diam}}(G)\leq \frac{n+17}{3}$$.

• The paper combines a Brooks-theorem-based face-coloring orientation with Menger-path decompositions to prove that every n-vertex planar triangulation has oriented diameter at most (2n+44)/5, improving the previous leading constant from 1/2 to 2/5.
• For 4-connected planar triangulations, the paper establishes the stronger bound (n+17)/3, confirming the conjectured n/3+O(1) behavior up to an additive constant and demonstrating how connectivity improves directed routing.
• The results leave the central 3-connected case open, with the main unresolved gap lying between the universal upper bound of approximately 2n/5 and nested-triangle lower bounds of approximately n/3.

Background and problem

The oriented diameter $\ordiam(G)$ of a connected bridgeless graph $G$ is the minimum diameter over all strong orientations of $G$. By Robbins' theorem, strong orientations exist exactly for bridgeless graphs, and the oriented-diameter problem asks how much routing efficiency must be sacrificed when every edge of a bidirectional network is assigned a single direction. Chvátal and Thomassen established the foundational results: deciding whether a graph admits an orientation of diameter two is NP-complete, and every bridgeless graph of diameter $d$ has oriented diameter at most $2d^2+2d$, with matching lower-bound examples up to a constant factor.

For planar triangulations specifically, Mondal, Parthiban, and Rajasingh proved an upper bound of $\frac n2 + O(\sqrt n)$ and constructed nested-triangle examples showing that some $n$-vertex planar triangulations have oriented diameter at least $n/3$. Ge, Liu, and Wang subsequently improved the general bound to $\lceil n/2\rceil$ for all but seven small exceptions among 2-connected near triangulations; this is sharp for near triangulations because maximal outerplanar graphs attain $\lceil n/2\rceil$. Wang showed that 5-connected triangulations admit $G$0. The paper under discussion narrows the gap between the lower bound $G$1 and the general upper bound $G$2, proving:

• Main result: every $G$3-vertex planar triangulation satisfies $G$4, improving the leading constant from $G$5 to $G$6.
• 4-connected case: every $G$7-vertex 4-connected planar triangulation satisfies $G$8, resolving the natural conjecture $G$9 in this setting.

The author formulates the target conjecture explicitly: there exists an absolute constant $G$0 such that every $G$1-vertex planar triangulation has $G$2. The conjecture remains open for connectivity three, and the paper does not claim to resolve it there.

A diameter-sensitive orientation via face coloring

The first tool is a linear relationship between ordinary and oriented diameter, valid for all non-complete triangulations: if $G$3 is a planar triangulation, then $G$4 admits a strong orientation $G$5 with $G$6 for all pairs, hence $G$7. The proof is short and elegant. The plane dual $G$8 is connected and cubic, so Brooks' theorem yields a proper 3-coloring of the faces of $G$9. Each edge is oriented so that its lower-colored incident face lies on its left. Faces colored 0 then bound counterclockwise directed triangles and faces colored 2 bound clockwise directed triangles, so every arc lies on a directed triangle. Any undirected shortest path can therefore be converted into a directed walk of at most twice its length by replacing each wrongly directed edge with a length-two detour through the third vertex of a directed triangle.

Combined with Menger's theorem (a $d$0-connected $d$1-vertex graph has diameter at most $d$2), this gives $d$3 for $d$4-connected triangulations, $d$5. For $d$6 this is weaker than the Ge–Liu–Wang bound; for $d$7 it is asymptotically equal to it; for $d$8 it matches Wang's asymptotic bound. The same face-coloring technique also produces a lemma for near triangulations: the interior edges can be oriented so that distances within components of $d$9 are at most doubled, and every interior vertex reaches the outer cycle (and is reached from it) within twice its undirected distance to $2d^2+2d$0. This lemma is essential for the 4-connected argument.

Triangulations with large diameter

The complementary estimate handles large ordinary diameter: every $2d^2+2d$1-vertex planar triangulation satisfies

$2d^2+2d$2

The proof exploits the fact that a triangulation on at least four vertices is 3-connected. Taking vertices $2d^2+2d$3 at distance $2d^2+2d$4, Menger's theorem supplies three internally disjoint $2d^2+2d$5-paths $2d^2+2d$6, each containing at least $2d^2+2d$7 vertices when $2d^2+2d$8. The exterior region bounded by $2d^2+2d$9 forms a near triangulation $\frac n2 + O(\sqrt n)$0 with at least $\frac n2 + O(\sqrt n)$1 vertices, orientable with diameter at most $\frac n2 + O(\sqrt n)$2 via the Ge–Liu–Wang theorem.

The technical core concerns the $\frac n2 + O(\sqrt n)$3-bridges attached to $\frac n2 + O(\sqrt n)$4: each non-edge bridge has a unique attachment $\frac n2 + O(\sqrt n)$5 on $\frac n2 + O(\sqrt n)$6 and attachments spanning a short interval of $\frac n2 + O(\sqrt n)$7 (of length at most two between extreme attachments). The author carves out subgraphs $\frac n2 + O(\sqrt n)$8 from these bridges—each a union of 2-connected near triangulations glued along at most one edge—and orients them using Lemma on special orientations, achieving diameter at most $\frac n2 + O(\sqrt n)$9 per piece with controlled distances to the attachment vertex. Gluing these orientations together (reversing pieces where necessary to make shared edges consistent) yields an orientation of a spanning subgraph $n$0 with $n$1; since every omitted vertex has at least two neighbors in $n$2, adding them back costs only 2, giving the stated bound.

Balancing the two estimates—$n$3 is maximized when $n$4, i.e., $n$5—yields the main theorem $n$6. The additive constant 44 is inherited directly from the threshold $n$7 at which the large-diameter regime activates; reducing this constant would require sharpening either the small-diameter analysis or the bridge-decomposition argument.

The 4-connected case

For 4-connected triangulations, both estimates improve. Four internally disjoint $n$8-paths allow the interior region $n$9 and exterior region $n/3$0 to be separated so that $n/3$1, and every vertex of $n/3$2 has a neighbor in $n/3$3. This last property lets the author extend an optimal orientation of $n/3$4 to the whole exterior subgraph $n/3$5 at a cost of only 4 in diameter, by closing directed triangles through each new vertex. In the interior, the face-coloring lemma orients $n/3$6, and 4-connectivity guarantees that each component $n/3$7 of $n/3$8 satisfies $n/3$9 for $\lceil n/2\rceil$0—four disjoint paths from $\lceil n/2\rceil$1 to distinct boundary vertices force one to be short. Combining these pieces gives

$\lceil n/2\rceil$2

Balancing against $\lceil n/2\rceil$3 yields $\lceil n/2\rceil$4, confirming the $\lceil n/2\rceil$5 conjecture for 4-connected triangulations with $\lceil n/2\rceil$6.

A remark extends the method to 5-connected triangulations: five disjoint paths improve the distance bounds inside $\lceil n/2\rceil$7 to $\lceil n/2\rceil$8, producing the diameter-sensitive bound $\lceil n/2\rceil$9 and, after balancing, $\lceil n/2\rceil$0—about $\lceil n/2\rceil$1, slightly better than the $\lceil n/2\rceil$2 general bound and consistent with Wang's earlier result for this connectivity class.

Limitations and open questions

Several caveats bear directly on the strength of the results. First, the general upper bound $\lceil n/2\rceil$3 still leaves a multiplicative gap against the nested-triangle lower bound of $\lceil n/2\rceil$4; the conjectured optimal leading constant $\lceil n/2\rceil$5 is established here only for 4-connected triangulations, and the 3-connected case remains open. Second, the additive constants are not optimized: the value 44 arises from the crude threshold $\lceil n/2\rceil$6 below which the Ge–Liu–Wang bound alone suffices, and the constant $\lceil n/2\rceil$7 in the 4-connected result is likewise an artifact of the balancing argument rather than believed tight. Third, the 5-connected improvement appears only as a sketched remark with the details declared identical to the 4-connected proof and omitted, so its stated bound $\lceil n/2\rceil$8 rests on an abbreviated verification. Finally, the lower-bound side is untouched: no new constructions are given, and whether the nested-triangle family is extremal for higher-connectivity classes is not addressed.

Conclusion

This paper improves the best universal upper bound on the oriented diameter of planar triangulations from $\lceil n/2\rceil$9 to $G$00, and proves the asymptotically correct $G$01 bound for the 4-connected class. The contribution is methodological as much as numerical: the combination of a Brooks-theorem-based face-coloring orientation (giving $G$02) with a Menger-path decomposition that converts large ordinary diameter into a savings term provides a clean template that already yields further gains at connectivity five. Closing the remaining gap between $G$03 and $G$04 for 3-connected triangulations is the central open problem left by this work.