On uniquely packable trees

An \(i\)-packing in a graph \(G\) is a set of vertices that are pairwise distance more than \(i\) apart. A \emph{packing colouring} of \(G\) is a partition \(X=\{X_{1},X_{2},\ldots,X_{k}\}\) of \(V(G)\) such that each colour class \(X_{i}\) is an \(i\)-packing. The minimum order \(k\) of a packing c...

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Bibliographic Details
Published inarXiv.org
Main Authors Alochukwu, A, Dorfling, M, Jonck, E
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.03.2024
Subjects
Apexes
Coloring
Graph theory
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Summary:An \(i\)-packing in a graph \(G\) is a set of vertices that are pairwise distance more than \(i\) apart. A \emph{packing colouring} of \(G\) is a partition \(X=\{X_{1},X_{2},\ldots,X_{k}\}\) 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 \(\chi_{\rho}(G)\). In this paper we investigate the existence of trees \(T\) for which there is only one packing colouring using \(\chi_\rho(T)\) colours. For the case \(\chi_\rho(T)=3\), we completely characterise all such trees. As a by-product we obtain sets of uniquely \(3\)-\(\chi_\rho\)-packable trees with monotone \(\chi_{\rho}\)-coloring and non-monotone \(\chi_{\rho}\)-coloring respectively.
ISSN:2331-8422