Vertex Ranking of Degenerate Graphs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917650033016832 |
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| author | Iacono, John Micek, Piotr Morin, Pat Reed, Bruce |
| author_facet | Iacono, John Micek, Piotr Morin, Pat Reed, Bruce |
| contents | An $\ell$-vertex-ranking of a graph $G$ is a colouring of the vertices of $G$ with integer colours so that in any connected subgraph $H$ of $G$ with diameter at most $\ell$, there is a vertex in $H$ whose colour is larger than that of every other vertex in $H$. The $\ell$-vertex-ranking number, $χ_{\ell-\mathrm{vr}}(G)$, of $G$ is the minimum integer $k$ such that $G$ has an $\ell$-vertex-ranking using $k$ colours. We prove that, for any fixed $d$ and $\ell$, every $d$-degenerate $n$-vertex graph $G$ satisfies $χ_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$ if $\ell$ is even and $χ_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$ if $\ell$ is odd. The case $\ell=2$ resolves (up to the $\log n$ factor) an open problem posed by \citet{karpas.neiman.ea:on} and the cases $\ell\in\{2,3\}$ are asymptotically optimal (up to the $\log n$ factor). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vertex Ranking of Degenerate Graphs Iacono, John Micek, Piotr Morin, Pat Reed, Bruce Combinatorics Discrete Mathematics An $\ell$-vertex-ranking of a graph $G$ is a colouring of the vertices of $G$ with integer colours so that in any connected subgraph $H$ of $G$ with diameter at most $\ell$, there is a vertex in $H$ whose colour is larger than that of every other vertex in $H$. The $\ell$-vertex-ranking number, $χ_{\ell-\mathrm{vr}}(G)$, of $G$ is the minimum integer $k$ such that $G$ has an $\ell$-vertex-ranking using $k$ colours. We prove that, for any fixed $d$ and $\ell$, every $d$-degenerate $n$-vertex graph $G$ satisfies $χ_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$ if $\ell$ is even and $χ_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$ if $\ell$ is odd. The case $\ell=2$ resolves (up to the $\log n$ factor) an open problem posed by \citet{karpas.neiman.ea:on} and the cases $\ell\in\{2,3\}$ are asymptotically optimal (up to the $\log n$ factor). |
| title | Vertex Ranking of Degenerate Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2404.16340 |