Proper Orientations of Planar Bipartite Graphs

TL;DR

This paper investigates proper orientations of planar bipartite graphs, demonstrating that certain orientations are possible with limited control, and establishes bounds on the proper orientation number for specific classes of bipartite graphs, including trees and 3-connected planar bipartite graphs.

Contribution

It shows that proper orientations can be achieved with limited edge control in bipartite graphs and provides bounds for the proper orientation number in trees and 3-connected planar bipartite graphs.

Findings

3-connected planar bipartite graphs have proper orientation number at most 6

Trees have proper orientation number at most 4 with a polynomial-time algorithm

Proper orientations are possible with control over a spanning tree and incident edges to a leaf

Abstract

An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrow{\chi}(G)$ 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. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that $\overrightarrow{\chi}(G) \leq 4$, when $G$ is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.