Proper Orientation Number of Triangle-free Bridgeless Outerplanar Graphs
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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 proper if neighbouring vertices have different in-degrees. The proper orientation number of a graph G, denoted by χ⃗(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 χ⃗(G)≤ c for every outerplanar graph G and showed that χ⃗(G)≤ 7 for every cactus G. We prove that χ⃗(G)≤ 3 if G is a triangle-free 2-connected outerplanar graph and χ⃗(G)≤ 4 if G is a triangle-free bridgeless outerplanar graph.
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