Abstract
AbstractAn orientation of is a digraph obtained from by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation proper if neighboring vertices have different in‐degrees. The proper orientation number of a graph , denoted by , is the minimum maximum in‐degree of a proper orientation of . Araujo et al asked whether there is a constant such that for every outerplanar graph and showed that for every cactus . We prove that if is a triangle‐free 2‐connected outerplanar graph and if is a triangle‐free bridgeless outerplanar graph.
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