On the proper orientation number of bipartite graphs

Abstract : An 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. We then prove that deciding whether − → χ (G) ≤ ∆(G) − 1 is an NP-complete problem. We also show that it is NP-complete to decide whether − → χ (G) ≤ 2, for planar subcubic graphs G. Moreover, we prove that it is NP-complete to decide whether − → χ (G) ≤ 3, for planar bipartite graphs G with maximum degree 5.
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Julio Araujo, Nathann Cohen, Susanna de Rezende, Frédéric Havet, Phablo Moura. On the proper orientation number of bipartite graphs. 9th International colloquium on graph theory and combinatorics, Jun 2014, Grenoble, France. ⟨hal-01076904⟩

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