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تسجيل مستخدم جديدProper 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 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.
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A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)rightarrow mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denote
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