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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.16573 |
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| _version_ | 1866911165716627456 |
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| author | Kittipassorn, Teeradej Suriya, Peerawit |
| author_facet | Kittipassorn, Teeradej Suriya, Peerawit |
| contents | For a non-decreasing sequence of positive integers $S = (a_1, a_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is the smallest positive integer $k$ such that the vertices can be colored with $k$ colors, where the distance between any two distinct vertices of color $i$ is greater than $a_i$. In this paper, we show that the $S$-packing chromatic number of the infinite diagonal grid $P_\infty \boxtimes P_\infty$ with $S = (1,6,6,\ldots)$ is $40$. This confirms a conjecture of the first author and Tiyajamorn. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $S$-packing coloring of the infinite diagonal grid with $S = (1,6,6,\ldots)$ Kittipassorn, Teeradej Suriya, Peerawit Combinatorics 05C15 (Primary) 05C35 (Secondary) For a non-decreasing sequence of positive integers $S = (a_1, a_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is the smallest positive integer $k$ such that the vertices can be colored with $k$ colors, where the distance between any two distinct vertices of color $i$ is greater than $a_i$. In this paper, we show that the $S$-packing chromatic number of the infinite diagonal grid $P_\infty \boxtimes P_\infty$ with $S = (1,6,6,\ldots)$ is $40$. This confirms a conjecture of the first author and Tiyajamorn. |
| title | The $S$-packing coloring of the infinite diagonal grid with $S = (1,6,6,\ldots)$ |
| topic | Combinatorics 05C15 (Primary) 05C35 (Secondary) |
| url | https://arxiv.org/abs/2509.16573 |