# Proper orientation number of triangle-free bridgeless outerplanar graphs

Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Yongtang Shi, Zhenyu Taoqiu

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An orientation of a graph 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 proper if neighbouring vertices have different in-degrees. The proper orientation number of a graph G, denoted by \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 \chi(G)< c for every outerplanar graph $G$ and showed that \chi}(G) <= 7$for every cactus G. We prove that \chi}(G) <=3 if G is a triangle-free 2-connected outerplanar graph and$\chi (G) <= 4\$ if G is a triangle-free bridgeless outerplanar graph.