### Abstract

Abstract We study vertex labelings φ:V → {0,1,2,...} of a graph G=(V,E) which assign nonnegative integers to the vertices and the restrictions depend on the distances in G. Fixing a positive integer d, the requirement is that if vertices u and v are at distance i apart (where 1≤i≤d), then |φ(u)-φ(v)|>d-i must hold. A corollary of the main result of this paper is an exact formula for the smallest possible value of max_{v∈V} φ(v) for trees whose internal vertices all have the same degree and all leaves are at distance d/2 from the central vertex (for d even) or at distance (d-1)/2 from the central edge (for d odd). The case of even diameter extends the main theorem of Li et al. (2010) on complete rooted trees with fixed down-degree and height.

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

Article number | 10086 |

Pages (from-to) | 1398-1406 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 338 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 6 2015 |

### Fingerprint

### Keywords

- Complete tree
- Graph coloring
- Multi-level distance labeling
- Radio labeling
- Radio number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*338*(8), 1398-1406. [10086]. https://doi.org/10.1016/j.disc.2015.02.016

**Distance-constrained labeling of complete trees.** / Halász, Veronika; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 338, no. 8, 10086, pp. 1398-1406. https://doi.org/10.1016/j.disc.2015.02.016

}

TY - JOUR

T1 - Distance-constrained labeling of complete trees

AU - Halász, Veronika

AU - Tuza, Z.

PY - 2015/8/6

Y1 - 2015/8/6

N2 - Abstract We study vertex labelings φ:V → {0,1,2,...} of a graph G=(V,E) which assign nonnegative integers to the vertices and the restrictions depend on the distances in G. Fixing a positive integer d, the requirement is that if vertices u and v are at distance i apart (where 1≤i≤d), then |φ(u)-φ(v)|>d-i must hold. A corollary of the main result of this paper is an exact formula for the smallest possible value of maxv∈V φ(v) for trees whose internal vertices all have the same degree and all leaves are at distance d/2 from the central vertex (for d even) or at distance (d-1)/2 from the central edge (for d odd). The case of even diameter extends the main theorem of Li et al. (2010) on complete rooted trees with fixed down-degree and height.

AB - Abstract We study vertex labelings φ:V → {0,1,2,...} of a graph G=(V,E) which assign nonnegative integers to the vertices and the restrictions depend on the distances in G. Fixing a positive integer d, the requirement is that if vertices u and v are at distance i apart (where 1≤i≤d), then |φ(u)-φ(v)|>d-i must hold. A corollary of the main result of this paper is an exact formula for the smallest possible value of maxv∈V φ(v) for trees whose internal vertices all have the same degree and all leaves are at distance d/2 from the central vertex (for d even) or at distance (d-1)/2 from the central edge (for d odd). The case of even diameter extends the main theorem of Li et al. (2010) on complete rooted trees with fixed down-degree and height.

KW - Complete tree

KW - Graph coloring

KW - Multi-level distance labeling

KW - Radio labeling

KW - Radio number

UR - http://www.scopus.com/inward/record.url?scp=84925423995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925423995&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2015.02.016

DO - 10.1016/j.disc.2015.02.016

M3 - Article

AN - SCOPUS:84925423995

VL - 338

SP - 1398

EP - 1406

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 8

M1 - 10086

ER -