On the proper orientation number of bipartite graphs

• Published in: Theoretical computer science Vol. 566; pp. 59 - 75
• Main Authors: Araujo, Julio, Cohen, Nathann, de Rezende, Susanna F., Havet, Frédéric, Moura, Phablo F.S.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 09.02.2015; Elsevier
• Subjects: Bipartite graph; Combinatorics; Computational Complexity; Computer Science; Graph coloring; Graph theory; Graphs; Mathematics; Orientation; Proper orientation; Trees; Proper orientation; Graph coloring; Bipartite graph
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2014.11.037

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. For each v∈V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u)≠dD−(v), for all uv∈E(G). The proper orientation number of a graph G, denoted by χ→(G), is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that χ→(G)≤(Δ(G)+Δ(G))/2+1 if G is a bipartite graph, and χ→(G)≤4 if G is a tree. It is well-known that χ→(G)≤Δ(G), for every graph G. However, we prove that deciding whether χ→(G)≤Δ(G)−1 is already an NP-complete problem on graphs with Δ(G)=k, for every k≥3. 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.