On λ-backbone coloring of cliques with tree backbones in linear time
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914748365275136 |
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| author | Michalik, Krzysztof Turowski, Krzysztof |
| author_facet | Michalik, Krzysztof Turowski, Krzysztof |
| contents | A $λ$-backbone coloring of a graph $G$ with its subgraph (also called a backbone) $H$ is a function $c \colon V(G) \rightarrow \{1,\dots, k\}$ ensuring that $c$ is a proper coloring of $G$ and for each $\{u,v\} \in E(H)$ it holds that $|c(u) - c(v)| \ge λ$. In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed $\max\{n, 2 λ\} + Δ(H)^2 \lceil\log{n} \rceil$. This result improves on the previously existing approximation algorithms as it is $(Δ(H)^2 \lceil\log{n} \rceil)$-absolutely approximate, i.e. with an additive error over the optimum. We also present an infinite family of trees $T$ with $Δ(T) = 3$ for which the coloring of cliques with backbones $T$ require to use at least $\max\{n, 2 λ\} + Ω(\log{n})$ colors for $λ$ close to $\frac{n}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_05772 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On λ-backbone coloring of cliques with tree backbones in linear time Michalik, Krzysztof Turowski, Krzysztof Discrete Mathematics Combinatorics 05C15 G.2.2; F.2.2 A $λ$-backbone coloring of a graph $G$ with its subgraph (also called a backbone) $H$ is a function $c \colon V(G) \rightarrow \{1,\dots, k\}$ ensuring that $c$ is a proper coloring of $G$ and for each $\{u,v\} \in E(H)$ it holds that $|c(u) - c(v)| \ge λ$. In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed $\max\{n, 2 λ\} + Δ(H)^2 \lceil\log{n} \rceil$. This result improves on the previously existing approximation algorithms as it is $(Δ(H)^2 \lceil\log{n} \rceil)$-absolutely approximate, i.e. with an additive error over the optimum. We also present an infinite family of trees $T$ with $Δ(T) = 3$ for which the coloring of cliques with backbones $T$ require to use at least $\max\{n, 2 λ\} + Ω(\log{n})$ colors for $λ$ close to $\frac{n}{2}$. |
| title | On λ-backbone coloring of cliques with tree backbones in linear time |
| topic | Discrete Mathematics Combinatorics 05C15 G.2.2; F.2.2 |
| url | https://arxiv.org/abs/2107.05772 |