Vertex Ranking of Degenerate Graphs

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, ${\chi }_{\ell -\mathrm{vr}}(G)$, 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).

15 pages, zero figures