### 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 |

DOIs | |

Publication status | Published - Feb 6 2000 |

### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*212*(1-2), 141-147. https://doi.org/10.1016/S0012-365X(99)00215-0