### Abstract

A (vertex) k-ranking of a graph G = (V, E) is a proper vertex coloring φ : V → {1,...,k} such that each path with endvertices of the same color i contains an internal vertex of color ≥ i + 1. In the on-line coloring algorithms, the vertices v_{1},...,v_{n} arrive one by one in an unrestricted order, and only the edges inside the set {v_{1},...,v_{i}} are known when the color of v_{i} has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log_{2} n)-ranking for the path with n ≥ 2 vertices, independently of the order in which the vertices are received.

Original language | English |
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Pages (from-to) | 141-147 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 212 |

Issue number | 1-2 |

Publication status | Published - Feb 6 2000 |

### Fingerprint

### Keywords

- On-line coloring algorithm
- Vertex coloring
- Vertex ranking

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*212*(1-2), 141-147.

**On-line rankings of graphs.** / Schiermeyer, I.; Tuza, Z.; Voigt, M.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 212, no. 1-2, pp. 141-147.

}

TY - JOUR

T1 - On-line rankings of graphs

AU - Schiermeyer, I.

AU - Tuza, Z.

AU - Voigt, M.

PY - 2000/2/6

Y1 - 2000/2/6

N2 - A (vertex) k-ranking of a graph G = (V, E) is a proper vertex coloring φ : V → {1,...,k} such that each path with endvertices of the same color i contains an internal vertex of color ≥ i + 1. In the on-line coloring algorithms, the vertices v1,...,vn arrive one by one in an unrestricted order, and only the edges inside the set {v1,...,vi} are known when the color of vi has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log2 n)-ranking for the path with n ≥ 2 vertices, independently of the order in which the vertices are received.

AB - A (vertex) k-ranking of a graph G = (V, E) is a proper vertex coloring φ : V → {1,...,k} such that each path with endvertices of the same color i contains an internal vertex of color ≥ i + 1. In the on-line coloring algorithms, the vertices v1,...,vn arrive one by one in an unrestricted order, and only the edges inside the set {v1,...,vi} are known when the color of vi has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log2 n)-ranking for the path with n ≥ 2 vertices, independently of the order in which the vertices are received.

KW - On-line coloring algorithm

KW - Vertex coloring

KW - Vertex ranking

UR - http://www.scopus.com/inward/record.url?scp=0042783049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042783049&partnerID=8YFLogxK

M3 - Article

VL - 212

SP - 141

EP - 147

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-2

ER -