Orientations of Graphs with Prescribed Weighted Out-Degrees

Michael Stiebitz, Z. Tuza, Margit Voigt

Research output: Contribution to journalArticle

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 (formula presented). Then the question is whether there exists an orientation D of G such that each vertex (formula presented) satisfies (formula presented). On one hand, this problem can be solved in polynomial time if g(v) = 1 for every vertex (formula presented). 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 (formula presented) and where we ask for an orientation D such that each vertex (formula presented) satisfies (formula presented), 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 (formula presented) for every vertex (formula presented).

Original language English 265-280 16 Graphs and Combinatorics 31 1 https://doi.org/10.1007/s00373-013-1382-0 Published - 2015

Fingerprint

Coloring
Graph in graph theory
Terminology
Vertex of a graph
Computational complexity
Polynomials
Weighted Graph
NP-complete problem
Choosability
List Coloring
Odd Cycle
Kernel Methods
Maximum Degree
Theorem
Complete Graph
Colouring
Connected graph
Polynomial time
kernel

Keywords

• Brooks’ theorem
• Chromatic number
• Kernels
• Orientations
• Vertex weighted graphs

ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Theoretical Computer Science

Cite this

Orientations of Graphs with Prescribed Weighted Out-Degrees. / Stiebitz, Michael; Tuza, Z.; Voigt, Margit.

In: Graphs and Combinatorics, Vol. 31, No. 1, 2015, p. 265-280.

Research output: Contribution to journalArticle

Stiebitz, Michael ; Tuza, Z. ; Voigt, Margit. / Orientations of Graphs with Prescribed Weighted Out-Degrees. In: Graphs and Combinatorics. 2015 ; Vol. 31, No. 1. pp. 265-280.
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