Zijian Ren | Searcharxiv

arXiv2026

The asymptotic maximum oriented diameter of graphs

The oriented diameter of a connected bridgeless graph is the minimum diameter of a strong orientation. Let $f(d)$ be the maximum oriented diameter among all such graphs of diameter $d$. In 1978, Chvátal and Thomassen proved $f(d)\le2d^2+2d$ and constructed graphs showing that any quadratic upper bound on $f(d)$ must have leading coefficient at least $1/2$. We prove that $f(d)\le \tfrac12d^2+7d$ for every integer $d\ge1$. This matches the leading coefficient of their lower bound and establishes $f(d)=\tfrac12d^2+O(d)$, determining the optimal quadratic coefficient. Our proof gives a polynomial-time algorithm that constructs a strong orientation satisfying the stated bound.

math.CO↗

arXiv2026

An improved upper bound for oriented diameter of graphs with diameter $4$

Let $f(d)$ denote the smallest integer such that every bridgeless graph of diameter $d$ admits a strong orientation with diameter at most $f(d)$. It is known that $f(2)=6$ and $f(3)=9$. For $d=4$, the classical bounds of Chvátal and Thomassen [JCTB, 1978] imply $12\le f(4)\le40$, and subsequent work reduced the upper bound to 21. Very recently, Lin, Wang and You further established the substantially stronger bound $f(4)\le18$. Pushing this bound below $18$ turns out to be considerably more difficult, since the remaining extremal configurations cannot be handled by existing techniques based on $R-S$ orientations and related local constructions. In this paper, we prove that $f(4)\le16$. Our approach is entirely different from previous ones. Instead of constructing a strong orientation directly, we develop a sequential orientation framework together with auxiliary distance functions and a potential-function analysis. This enables us to control directed distances globally while avoiding the intricate case analysis required by earlier methods. We believe that the framework introduced here may be useful for studying oriented diameter problems of larger diameter.

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