Grundy Packing Coloring of Graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912281831407616 |
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| author | Gözüpek, Didem Peterin, Iztok |
| author_facet | Gözüpek, Didem Peterin, Iztok |
| contents | A map $c:V(G)\rightarrow\{1,\dots,k\}$ of a graph $G$ is a packing $k$-coloring if every two different vertices of the same color $i\in \{1,\dots,k\}$ are at distance more than $i$. The packing chromatic number $χ_ρ(G)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of \textit{Grundy packing chromatic number}, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of $G$. We then define the Grundy packing chromatic number $Γ_ρ(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of $Γ_ρ(G)$, provide results on the complexity of the problem as well as bounds and some exact results for $Γ_ρ(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grundy Packing Coloring of Graphs Gözüpek, Didem Peterin, Iztok Combinatorics 05C15, 05C12, 05C69 A map $c:V(G)\rightarrow\{1,\dots,k\}$ of a graph $G$ is a packing $k$-coloring if every two different vertices of the same color $i\in \{1,\dots,k\}$ are at distance more than $i$. The packing chromatic number $χ_ρ(G)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of \textit{Grundy packing chromatic number}, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of $G$. We then define the Grundy packing chromatic number $Γ_ρ(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of $Γ_ρ(G)$, provide results on the complexity of the problem as well as bounds and some exact results for $Γ_ρ(G)$. |
| title | Grundy Packing Coloring of Graphs |
| topic | Combinatorics 05C15, 05C12, 05C69 |
| url | https://arxiv.org/abs/2409.00697 |