Abstract: The oriented diameter of an undirected graph G, denoted by \begin{document}$ \textrm{diam}(\overrightarrow{G}) $\end{document}, is defined as the minimum diameter of any strong orientation of G. In this paper, we prove that the oriented diameter of a bridgeless \begin{document}$ C_{4} $\end{document}-free graph is at most \begin{document}$ \frac{37n}{\delta ^2-2\lfloor \delta /2\rfloor +1}+19 $\end{document}. We also consider some upper bounds of \begin{document}$ \textrm{diam}(\overrightarrow{G}) $\end{document} in terms of girth g, minimum degree \begin{document}$ \delta $\end{document} and maximum degree \begin{document}$ \Delta $\end{document} and give some upper bounds of \begin{document}$ \textrm{diam}(\overrightarrow{G}) $\end{document} in bridgeless \begin{document}$ (C_{4},C_{5}) $\end{document}-free graphs.

Key words: Orientation, Diameter, Minimum degree, Maximum degree, Girth, Packing, Bridgeless graph