Fast algorithm for $S$-packing coloring of Halin graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908735104876544 |
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| author | Zhang, Xin Zou, Dezhi |
| author_facet | Zhang, Xin Zou, Dezhi |
| contents | Motivated by frequency assignment problems in wireless broadcast networks, Goddard, Hedetniemi, Hedetniemi, Harris, and Rall introduced the notion of $S$-packing coloring in 2008. Given a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $k$ subsets $\{V_1, V_2, \ldots, V_k\}$ such that for each $1 \leq i \leq k$, the distance between any two distinct vertices $u, v \in V_i$ is at least $s_i + 1$. In this paper, we study the $S$-packing coloring problem for Halin graphs with maximum degree $Δ\leq 5$. Specifically, we present a linear-time algorithm that constructs a $(1,1,2,2,2)$-packing coloring for any Halin graph satisfying $Δ\leq 5$. It is worth noting that there are Halin graphs that are not $(1,2,2,2)$-packing colorable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast algorithm for $S$-packing coloring of Halin graphs Zhang, Xin Zou, Dezhi Combinatorics Discrete Mathematics Motivated by frequency assignment problems in wireless broadcast networks, Goddard, Hedetniemi, Hedetniemi, Harris, and Rall introduced the notion of $S$-packing coloring in 2008. Given a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $k$ subsets $\{V_1, V_2, \ldots, V_k\}$ such that for each $1 \leq i \leq k$, the distance between any two distinct vertices $u, v \in V_i$ is at least $s_i + 1$. In this paper, we study the $S$-packing coloring problem for Halin graphs with maximum degree $Δ\leq 5$. Specifically, we present a linear-time algorithm that constructs a $(1,1,2,2,2)$-packing coloring for any Halin graph satisfying $Δ\leq 5$. It is worth noting that there are Halin graphs that are not $(1,2,2,2)$-packing colorable. |
| title | Fast algorithm for $S$-packing coloring of Halin graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2512.22809 |