Kaydedildi:
| Yazar: | |
|---|---|
| Materyal Türü: | Preprint |
| Baskı/Yayın Bilgisi: |
2024
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| Konular: | |
| Online Erişim: | https://arxiv.org/abs/2406.00643 |
| Etiketler: |
Etiketle
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İçindekiler:
- The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $Γ(G)$ (or $χ_{_{\sf FF}}(G)$), is the maximum number of colors used by a First-Fit (greedy) coloring of $G$. To determine $Γ(G)$ is NP-complete for various classes of graphs. Also there exists a constant $c>0$ such that the Grundy number is hard to approximate within the ratio $c$. We first obtain an $\mathcal{O}(VE)$ algorithm to determine the Grundy number of block graphs i.e. graphs in which every biconnected component is complete subgraph. We prove that the Grundy number of a general graph $G$ with cut-vertices is upper bounded by the Grundy number of a block graph corresponding to $G$. This provides a reasonable upper bound for the Grundy number of graphs with cut-vertices. Next, define $Δ_2(G)={\max}_{u\in G}~ {\max}_{v\in N(u):d(v)\leq d(u)} d(v)$. We obtain an $\mathcal{O}(VE)$ algorithm to determine $Γ(G)$ for graphs $G$ whose girth $g$ is at least $2Δ_2(G)+1$. This algorithm provides a polynomial time approximation algorithm within ratio $\min \{1, (g+1)/(2Δ_2(G)+2)\}$ for $Γ(G)$ of general graphs $G$ with girth $g$.