An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chv\'atal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which outperforms the former upper bound for all values of $d$ greater than or equal to $8$. For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of $9$ and $11$ respectively. For the family of $2$-edge connected graphs of diameter $4$, the bounds provided by Chv\'atal and Thomassen are $12$ and $40$ and no better bounds were known. By extending the method we used for diameter $d$ graphs, along with an asymmetric extension of a technique used by Chv\'atal and Thomassen, we have improved this upper bound to $21$.

中文翻译：

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Chv\'atal 和 Thomassen 的定向直径上界的改进

无向图 $G$ 的方向是为 $G$ 的每条边指定一个方向。图的定向直径$G$是$G$所有方向中最小的直径。图族$\mathscr{F}$的最大定向直径是$\mathscr{F}$中所有图的最大定向直径。Chv\'atal 和 Thomassen [JCTB, 1978] 给出了 $\frac{1}{2}d^2+d$ 的下限和 $2d^2+2d$ 的上限，用于最大定向直径直径为 $d$ 的 $2$-边连通图族。我们将此上限改进为 $1.373 d^2 + 6.971d-1 $，对于大于或等于 $8$ 的所有 $d$ 值，该上限优于之前的上限。对于直径为 $3$、Kwok、Liu 和 West [JCTB，2010] 分别获得了 9 美元和 11 美元的改进下限和上限。对于直径为 $4$ 的 $2$-边连通图族，Chv\'atal 和 Thomassen 提供的边界为 $12$ 和 $40$，并且没有更好的边界已知。通过扩展我们用于直径 $d$ 图形的方法，以及 Chv\'atal 和 Thomassen 使用的技术的非对称扩展，我们将这个上限提高到 $21$。