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 $${f, g : V(G) \to \mathbb{N}}$$ f , g : V ( G ) ? N . Then the question is whether there exists an orientation D of G such that each vertex $${v \in V(G)}$$ v ? V ( G ) satisfies $${\sum_{u \in N_D^{+}(v)}g(u) \leq f(v)}$$ ? u ? N D + ( v ) g ( u ) ≤ f ( v ) . On one hand, this problem can be solved in polynomial time if g(v) = 1 for every vertex $${v \in V(G)}$$ v ? 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 $${h : E(G) \to \mathbb{N}}$$ h : E ( G ) ? N and where we ask for an orientation D such that each vertex $${v \in V(G)}$$ v ? V ( G ) satisfies $${\sum_{e \in E_D^{+}(v)}h(e) \leq f(v)}$$ ? e ? E D + ( v ) h ( e ) ≤ 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 $${d_{D}^{+}(v) \leq d_G(v)-1}$$ d D + ( v ) ≤ d G ( v ) - 1 for every vertex $${v \in V(G)}$$ v ? V ( G ) .