Vertex Ranking of Degenerate Graphs
<|reference_start|>Vertex Ranking of Degenerate Graphs: 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, $\chi_{\ell-\mathrm{vr}}(G)$, 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 $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$ if $\ell$ is even and $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$ 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).<|reference_end|>
@article{iacono2024vertex,
    title={Vertex Ranking of Degenerate Graphs},
    author={John Iacono, Piotr Micek, Pat Morin, and Bruce Reed},
    journal={arXiv preprint arXiv:2404.16340},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.16340},
    primaryClass={math.CO cs.DM}
}