On Grundy and b-chromatic number of some families of graphs: a comparative study
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| Format: | Preprint |
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2020
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| _version_ | 1866911788097863680 |
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| author | Masih, Zoya Zaker, Manouchehr |
| author_facet | Masih, Zoya Zaker, Manouchehr |
| contents | The Grundy and the {\rm b}-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph $G$, denoted by $Γ(G)$ is the worst case behavior of greedy (First-Fit) coloring procedure for $G$ and the {\rm b}-chromatic number ${\rm{b}}(G)$ is the maximum number of colors used in any color-dominating coloring of $G$. Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence $\{G_n\}_{n\geq 1}$ with limited {\rm b}-chromatic number but $Γ(G_n)\rightarrow \infty$. We obtain families of graphs $\mathcal{F}$ such that for some adequate function $f(.)$, $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. This verifies a previous conjecture for these families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_10070 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Grundy and b-chromatic number of some families of graphs: a comparative study Masih, Zoya Zaker, Manouchehr Combinatorics The Grundy and the {\rm b}-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph $G$, denoted by $Γ(G)$ is the worst case behavior of greedy (First-Fit) coloring procedure for $G$ and the {\rm b}-chromatic number ${\rm{b}}(G)$ is the maximum number of colors used in any color-dominating coloring of $G$. Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence $\{G_n\}_{n\geq 1}$ with limited {\rm b}-chromatic number but $Γ(G_n)\rightarrow \infty$. We obtain families of graphs $\mathcal{F}$ such that for some adequate function $f(.)$, $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. This verifies a previous conjecture for these families. |
| title | On Grundy and b-chromatic number of some families of graphs: a comparative study |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2012.10070 |