### Abstract

Let G=(V,E) be a simple graph, and for all v∈V, let L(v) be a list of colors assigned to v. We shall assume throughout that the colors are natural numbers. For nonnegative integers d,s, define χℓd,s(G) to be the smallest integer k such that for every list assignment with |L(v)|=k for all v∈V one can choose a color c(v)∈L(v) for every vertex in such a way that |c(v)-c(w)|≥d for all vw∈E and |c(v)-c(w)|≥s for all pairs v,w of vertices having distance 2 in G. For a given list assignment such a coloring c is called an L(d,s)-list labeling. We prove a general bound for χℓd,s(G) depending on the maximum degree of G. Furthermore, we study this parameter for trees, and also for the particular classes of paths and stars. Polynomial algorithms are designed for deciding whether a given list assignment admits an L(d,s)-list labeling on paths (for a given s unrestricted) and on trees (for s=1).

Original language | English |
---|---|

Pages (from-to) | 92-98 |

Number of pages | 7 |

Journal | Theoretical Computer Science |

Volume | 349 |

Issue number | 1 |

DOIs | |

Publication status | Published - dec. 12 2005 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*349*(1), 92-98. https://doi.org/10.1016/j.tcs.2005.09.032

**List version of L (d,s)-labelings.** / Kohl, Anja; Schreyer, Jens; Tuza, Z.; Voigt, Margit.

Research output: Article

*Theoretical Computer Science*, vol. 349, no. 1, pp. 92-98. https://doi.org/10.1016/j.tcs.2005.09.032

}

TY - JOUR

T1 - List version of L (d,s)-labelings

AU - Kohl, Anja

AU - Schreyer, Jens

AU - Tuza, Z.

AU - Voigt, Margit

PY - 2005/12/12

Y1 - 2005/12/12

N2 - Let G=(V,E) be a simple graph, and for all v∈V, let L(v) be a list of colors assigned to v. We shall assume throughout that the colors are natural numbers. For nonnegative integers d,s, define χℓd,s(G) to be the smallest integer k such that for every list assignment with |L(v)|=k for all v∈V one can choose a color c(v)∈L(v) for every vertex in such a way that |c(v)-c(w)|≥d for all vw∈E and |c(v)-c(w)|≥s for all pairs v,w of vertices having distance 2 in G. For a given list assignment such a coloring c is called an L(d,s)-list labeling. We prove a general bound for χℓd,s(G) depending on the maximum degree of G. Furthermore, we study this parameter for trees, and also for the particular classes of paths and stars. Polynomial algorithms are designed for deciding whether a given list assignment admits an L(d,s)-list labeling on paths (for a given s unrestricted) and on trees (for s=1).

AB - Let G=(V,E) be a simple graph, and for all v∈V, let L(v) be a list of colors assigned to v. We shall assume throughout that the colors are natural numbers. For nonnegative integers d,s, define χℓd,s(G) to be the smallest integer k such that for every list assignment with |L(v)|=k for all v∈V one can choose a color c(v)∈L(v) for every vertex in such a way that |c(v)-c(w)|≥d for all vw∈E and |c(v)-c(w)|≥s for all pairs v,w of vertices having distance 2 in G. For a given list assignment such a coloring c is called an L(d,s)-list labeling. We prove a general bound for χℓd,s(G) depending on the maximum degree of G. Furthermore, we study this parameter for trees, and also for the particular classes of paths and stars. Polynomial algorithms are designed for deciding whether a given list assignment admits an L(d,s)-list labeling on paths (for a given s unrestricted) and on trees (for s=1).

KW - Channel assignment problem

KW - L(d,s)-labeling

KW - List coloring

UR - http://www.scopus.com/inward/record.url?scp=27844593163&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27844593163&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2005.09.032

DO - 10.1016/j.tcs.2005.09.032

M3 - Article

AN - SCOPUS:27844593163

VL - 349

SP - 92

EP - 98

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1

ER -