Vertex Ranking of Degenerate Graphs

TL;DR

This paper establishes bounds on the vertex-ranking number for degenerate graphs, resolving an open problem for the case 62 and providing asymptotically optimal results for small 6 and 63.

Contribution

It provides new bounds on the 6-vertex-ranking number for degenerate graphs, solving an open problem for 62 and improving understanding of graph coloring complexity.

Findings

For fixed d and 6, the 6-vertex-ranking number is bounded by O(n^{1-2/(6+1)} 7 log n) if 6 is even.

For odd 6, the bound is O(n^{1-2/6} 7 log n).

The case 62 resolves an open problem up to a 7 log n factor.

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).