An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$ such that $\vec{\chi}(G)\leq c$ for every outerplanar graph $G$ and showed that $\vec{\chi}(G)\leq 7$ for every cactus $G.$ We prove that $\vec{\chi}(G)\leq 3$ if $G$ is a triangle-free $2$-connected outerplanar graph and $\vec{\chi}(G)\leq 4$ if $G$ is a triangle-free bridgeless outerplanar graph.

中文翻译：

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无三角无桥外平面图的正确取向数

$G$ 的方向是通过将每条边替换为具有相同端点的两个可能弧中的一个来从 $G$ 获得的有向图。如果相邻顶点具有不同的入度，我们称方向为 \emph{proper}。图 $G$ 的正确方向数，表示为 $\vec{\chi}(G)$，是 G. Araujo 等人的正确方向的最小最大入度。(Theor. Comput. Sci. 639 (2016) 14--25) 询问是否存在常数 $c$ 使得 $\vec{\chi}(G)\leq c$ 对于每个外平面图 $G$ 并显示$\vec{\chi}(G)\leq 7$ 对于每个仙人掌 $G.$ 我们证明 $\vec{\chi}(G)\leq 3$ 如果 $G$ 是无三角形的 $2$ -连通外平面图和 $\vec{\chi}(G)\leq 4$ 如果 $G$ 是无三角形无桥外平面图。