On uniquely packable trees

Abstract

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.

Figures (8)

• Figure 1: The three graphs from which all uniquely 3-$\chi_\rho$-packable trees are constructed.
• Figure 2: The seven operations used to construct all uniquely 3-$\chi_\rho$-packable trees.
• Figure 3: The trees $F_1$, $F_2$ and $F_3$ from which all uniquely 3-$\chi_\rho$-packable trees are constructed.
• Figure 4: The tree $T_k$ from which a set of uniquely $3$-$\chi_\rho$-packable trees with non-monotone colouring is constructed.
• Figure 5: The tree $T_{\ell,k}$ from which a set of uniquely 3-$\chi_\rho$-packable trees with monotone colouring is constructed.

Theorems & Definitions (16)

• Claim 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof