# Distance-constrained labeling of complete trees

Veronika Halász, Z. Tuza

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 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.

Original language English 10086 1398-1406 9 Discrete Mathematics 338 8 https://doi.org/10.1016/j.disc.2015.02.016 Published - Aug 6 2015

### Fingerprint

Labeling
Vertex Labeling
Integer
Rooted Trees
Assign
Corollary
Leaves
Odd
Non-negative
Restriction
Internal
Requirements
Graph in graph theory
Vertex of a graph
Theorem

### Keywords

• Complete tree
• Graph coloring
• Multi-level distance labeling

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Theoretical Computer Science

### Cite this

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

In: Discrete Mathematics, Vol. 338, No. 8, 10086, 06.08.2015, p. 1398-1406.

Research output: Contribution to journalArticle

Halász, Veronika ; Tuza, Z. / Distance-constrained labeling of complete trees. In: Discrete Mathematics. 2015 ; Vol. 338, No. 8. pp. 1398-1406.
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