Orientations of Graphs with Prescribed Weighted Out-Degrees

Abstract

If we want to apply Galvin’s kernel method to show that a graph G satisfies a certain coloring property, we have to find an appropriate orientation of G. This motivated us to investigate the complexity of the following orientation problem. The input is a graph G and two vertex functions ${f, g : V(G) \to \mathbb{N}}$. Then the question is whether there exists an orientation D of G such that each vertex ${v \in V(G)}$ satisfies ${\sum_{u \in N_D^{+}(v)}g(u) \leq f(v)}$. On one hand, this problem can be solved in polynomial time if g(v) = 1 for every vertex ${v \in V(G)}$. On the other hand, as proved in this paper, the problem is NP-complete even if we restrict it to graphs which are bipartite, planar and of maximum degree at most 3 and to functions f, g where the permitted values are 1 and 2, only. We also show that the analogous problem, where we replace g by an edge function ${h : E(G) \to \mathbb{N}}$ and where we ask for an orientation D such that each vertex ${v \in V(G)}$ satisfies ${\sum_{e \in E_D^{+}(v)}h(e) \leq f(v)}$, is NP-complete, too. Furthermore, we prove some new results related to the (f, g)-choosability problem, or in our terminology, to the list-coloring problem of weighted graphs. In particular, we use Galvin’s theorem to prove a generalization of Brooks’s theorem for weighted graphs. We show that if a connected graph G has a block which is neither a complete graph nor an odd cycle, then G has a kernel perfect super-orientation D such that ${d_{D}^{+}(v) \leq d_G(v)-1}$ for every vertex ${v \in V(G)}$.