Vertex Ranking of Degenerate Graphs
John Iacono, Piotr Micek, Pat Morin, Bruce Reed
TL;DR
The paper addresses the problem of bounding the $\ell$-vertex-ranking number for $d$-degenerate graphs, achieving parity-dependent upper bounds: $\chi_{\ell-\mathrm{vr}}(G) = O(n^{1-2/(\ell+1)}\log n)$ for even $\ell$ and $O(n^{1-2/\ell}\log n)$ for odd $\ell$. The authors introduce a two-phase probabilistic colouring strategy built on a Cairns-type endpoint mapping for $\ell$-paths, together with a degeneracy bound for $G^{\ell}$ and a careful tail analysis using Bernstein concentration. A key outcome is that the case $\ell=2$ (and hence $\ell=3$) matches known lower bounds up to a $\log n$ factor, and the results extend to handling maximum degree $\Delta$ through a decomposition argument. The work advances sparsity-related coloring theory by providing tight (up to logs) bounds for the centered vertex-ranking problem on sparse graphs and illustrates a powerful blend of combinatorial planning and probabilistic concentration techniques. These insights contribute to a deeper understanding of hierarchical colorings relevant to graph depth and sparsity.
Abstract
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).