On uniquely packable trees

An i-packing in a graph G is a set of vertices that are pairwise at distance more than i. A packing colouring of G is a partition X = {X 1 , X 2 , . . . , X k } of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number...

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Bibliographic Details
Published inQuaestiones mathematicae Vol. 47; no. 7; pp. 1353 - 1368
Main Authors Alochukwu, A., Dorfling, M., Jonck, E.
Format Journal Article
LanguageEnglish
Published Grahamstown Taylor & Francis 12.07.2024
Taylor & Francis Ltd
Subjects
Apexes
broadcast
Coloring
Colouring
Graph theory
monotone colouring
packing
packing chromatic number
tree
uniquely colourable
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Summary:An i-packing in a graph G is a set of vertices that are pairwise at distance more than i. A packing colouring of G is a partition X = {X 1 , X 2 , . . . , X k } of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χ ρ (G). In this paper we investigate the existence of trees T for which there is only one packing colouring using χ ρ (T) colours. For the case χ ρ (T) = 3, we completely characterise all such trees. As a by-product we obtain sets of uniquely 3-χ ρ -packable trees with monotone χ ρ -colouring and non-monotone χ ρ -colouring respectively.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2024.2321259