On uniquely packable trees
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910363188985856 |
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| author | Alochukwu, A. Dorfling, M. Jonck, E. |
| author_facet | Alochukwu, A. Dorfling, M. Jonck, E. |
| contents | 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 $χ_ρ(G)$. 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 $χ_ρ$-coloring and non-monotone $χ_ρ$-coloring respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_10889 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On uniquely packable trees Alochukwu, A. Dorfling, M. Jonck, E. Combinatorics 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 $χ_ρ(G)$. 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 $χ_ρ$-coloring and non-monotone $χ_ρ$-coloring respectively. |
| title | On uniquely packable trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2304.10889 |