$S$-packing colorings of distance graphs with distance sets of cardinality $2$
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| Format: | Preprint |
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2024
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| _version_ | 1866917678027898880 |
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| author | Brešar, Boštjan Ferme, Jasmina Holub, Přemysl Jakovac, Marko Melicharová, Petra |
| author_facet | Brešar, Boštjan Ferme, Jasmina Holub, Přemysl Jakovac, Marko Melicharová, Petra |
| contents | For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every $i\in\{1,\ldots,\ell\}$, is called an $S$-packing $\ell$-coloring of $G$. The minimum $\ell$ for which $G$ admits an $S$-packing $\ell$-coloring is called the $S$-packing chromatic number of $G$, denoted by $χ_S(G)$. In this paper, we consider $S$-packing colorings of distance graphs $G(\mathbb{Z},\{k,t\})$, where $k$ and $t$ are positive integers, which are the graphs whose vertex set is $\mathbb{Z}$, and two vertices $x,y\in \mathbb{Z}$ are adjacent whenever $|x-y|\in\{k,t\}$. We complement partial results from two earlier papers, thus determining all values of $χ_S(G(\mathbb{Z},\{k,t\}))$ when $S$ is any sequence with $s_i\le 2$ for all $i$. In particular, if $S=(1,1,2,2,\ldots)$, then the $S$-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while if $S=(1,2,2,\ldots)$, then the $S$-packing chromatic number is $5$, unless $\{k,t\}=\{2,3\}$ when it is $6$; when $S=(2,2,2,\ldots)$, the corresponding formula is more complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $S$-packing colorings of distance graphs with distance sets of cardinality $2$ Brešar, Boštjan Ferme, Jasmina Holub, Přemysl Jakovac, Marko Melicharová, Petra Combinatorics 05C15, 05C12 For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every $i\in\{1,\ldots,\ell\}$, is called an $S$-packing $\ell$-coloring of $G$. The minimum $\ell$ for which $G$ admits an $S$-packing $\ell$-coloring is called the $S$-packing chromatic number of $G$, denoted by $χ_S(G)$. In this paper, we consider $S$-packing colorings of distance graphs $G(\mathbb{Z},\{k,t\})$, where $k$ and $t$ are positive integers, which are the graphs whose vertex set is $\mathbb{Z}$, and two vertices $x,y\in \mathbb{Z}$ are adjacent whenever $|x-y|\in\{k,t\}$. We complement partial results from two earlier papers, thus determining all values of $χ_S(G(\mathbb{Z},\{k,t\}))$ when $S$ is any sequence with $s_i\le 2$ for all $i$. In particular, if $S=(1,1,2,2,\ldots)$, then the $S$-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while if $S=(1,2,2,\ldots)$, then the $S$-packing chromatic number is $5$, unless $\{k,t\}=\{2,3\}$ when it is $6$; when $S=(2,2,2,\ldots)$, the corresponding formula is more complex. |
| title | $S$-packing colorings of distance graphs with distance sets of cardinality $2$ |
| topic | Combinatorics 05C15, 05C12 |
| url | https://arxiv.org/abs/2405.18904 |