### 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 |
---|---|

Pages (from-to) | 265-280 |

Number of pages | 16 |

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*31*(1), 265-280. https://doi.org/10.1007/s00373-013-1382-0

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

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 31, no. 1, pp. 265-280. https://doi.org/10.1007/s00373-013-1382-0

}

TY - JOUR

T1 - Orientations of Graphs with Prescribed Weighted Out-Degrees

AU - Stiebitz, Michael

AU - Tuza, Z.

AU - Voigt, Margit

PY - 2015

Y1 - 2015

N2 - 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).

AB - 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).

KW - Brooks’ theorem

KW - Chromatic number

KW - Kernels

KW - Orientations

KW - Vertex weighted graphs

UR - http://www.scopus.com/inward/record.url?scp=84943589023&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84943589023&partnerID=8YFLogxK

U2 - 10.1007/s00373-013-1382-0

DO - 10.1007/s00373-013-1382-0

M3 - Article

VL - 31

SP - 265

EP - 280

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -