Proper Orientations of Planar Bipartite Graphs

Abstract

An orientation of a graph G is proper if any two adjacent vertices have different indegrees. The proper orientation number − → χ (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 − → χ (G) ≤ 4, when G is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.

Weighted proper orientations of trees and graphs of bounded treewidth

“…Interestingly, our pseudo-polynomial algorithm uses as a black box a subroutine to solve an appropriately defined Subset Sum instance. Another ingredient of this algorithm is a combinatorial lemma stating that there always exists a weighted proper orientation D of a tree T such that d − D (u) ≤ 4, for every u ∈ V (T ); this generalizes the result of Araújo et al [4] and Knox et al [16] for the unweighted version.…”

“…They proved that the problem of determining the proper orientation number of a graph remains NP-hard for subclasses of planar graphs that are also bipartite and of bounded degree. In the same paper, they proved that the proper orientation number of any tree is at most 4; Knox et al [16] provided a shorter proof of the same result. In another paper, Araújo et al [5] proved that the proper orientation number of cacti is at most 7, and that this bound is tight.…”

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3

“…The proper orientation number of G, denoted by − → χ (G), is the minimum integer k such that G admits a proper k-orientation. Proper orientation number is defined by Ahadi and Dehghan [1], and see the related research [2,3,6]. Knox et al [6] showed the following.…”

Proper orientation number of triangle‐free bridgeless outerplanar graphs

“…This makes it difficult to prove upper bounds on the parameter even for relatively narrow classes of graphs. Araujo et al [3] asked whether there is a constant c such that χ(G) ≤ c for every outerplanar graph G. They proved that for any cactus G (where every 2-connected component is either an edge or a cycle), we have χ(G) ≤ 7 and that for any tree T, χ(T ) ≤ 4 (see also [7] for a short algorithmic proof).…”

“…In particular, Knox el al. [7] proved that χ(G) ≤ 5 for a 3-connected planar bipartite graph G and Noguci [8] showed that χ(G) ≤ 3 for any bipartite planar graph with δ(G) ≥ 3.…”