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On the proper orientation number of bipartite graphs

Abstract : An {\it orientation} of a graph $G$ is a digraph $D$ obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each $v \in V(G)$, the \emph{indegree} of $v$ in $D$, denoted by $d^-_D(v)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is \emph{proper} if $d^-_D(u)\neq d^-_D(v)$, for all $uv\in E(G)$. The \emph{proper orientation number} of a graph $G$, denoted by $po(G)$, is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that $po(G) \leq \left(\Delta(G) + \sqrt{\Delta(G)}\right)/2 + 1$ if $G$ is a bipartite graph, and $po(G)\leq 4$ if $G$ is a tree. % Moreover, we show that deciding whether the proper orientation number is at most~2 and at most~3 % is an $NP$-complete problem for planar subcubic graphs and planar bipartite graphs, respectively. It is well-known that $po(G)\leq \Delta(G)$, for every graph $G$. However, we prove that deciding whether $po(G)\leq \Delta(G)-1$ is already an $NP$-complete problem on graphs with $\Delta(G) = k$, for every $k \geq 3$. We also show that it is $NP$-complete to decide whether $po(G)\leq 2$, for planar \emph{subcubic} graphs $G$. Moreover, we prove that it is $NP$-complete to decide whether $po(G)\leq 3$, for planar bipartite graphs $G$ with maximum degree 5.
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Contributor : Nathann Cohen <>
Submitted on : Friday, January 9, 2015 - 9:36:42 AM
Last modification on : Friday, April 30, 2021 - 9:58:17 AM

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Julio Araujo, Nathann Cohen, Susanna F. de Rezende, Frédéric Havet, Phablo Moura. On the proper orientation number of bipartite graphs. Theoretical Computer Science, Elsevier, 2015, 566, pp.59-75. ⟨10.1016/j.tcs.2014.11.037⟩. ⟨hal-01101578⟩



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