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).

翻译：设$G$是一个图，图$G$的$\ell$-顶点排序是指用整数对$G$的顶点进行着色，使得在$G$中任意直径不超过$\ell$的连通子图$H$中，存在$H$中的一个顶点，其颜色严格大于$H$中所有其他顶点的颜色。图$G$的$\ell$-顶点排序数$\chi_{\ell-\mathrm{vr}}(G)$是使得$G$存在使用$k$种颜色的$\ell$-顶点排序的最小整数$k$。我们证明，对于任意固定的$d$和$\ell$，每个$d$-退化$n$顶点图$G$满足：当$\ell$为偶数时，$\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$；当$\ell$为奇数时，$\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$。其中$\ell=2$的情形（在$\log n$因子范围内）解决了由\citet{karpas.neiman.ea:on}提出的一个开放问题，而$\ell\in\{2,3\}$的情形（在$\log n$因子范围内）是渐近最优的。