Proper Orientations of Planar Bipartite Graphs

Abstract

An orientation of a graph G is proper if any two adjacent vertices have different indegrees so that the values of the indegrees define a coloring of G. The proper orientation number $${{\mathrm{\overrightarrow{\chi }}}}(G)$$ of a graph G is the minimum of the maximum indegree, taken over all proper orientations of G. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. By using the same technique, we prove that 3-connected planar bipartite graphs have proper orientation number at most 5. This statement is the most general result in support of the open problem of whether all planar graphs have bounded proper orientation number. Additionally, we give a short proof that $${{\mathrm{\overrightarrow{\chi }}}}(G) \le 4$$ , when G is a tree and this proof leads to a polynomial-time algorithm to properly orient trees within this bound.