• Corpus ID: 220496342

Asymptotically Optimal Vertex Ranking of Planar Graphs

  title={Asymptotically Optimal Vertex Ranking of Planar Graphs},
  author={Prosenjit Bose and Vida Dujmovi'c and Mehrnoosh Javarsineh and Pat Morin},
A (vertex) $\ell$-ranking is a labelling $\varphi:V(G)\to\mathbb{N}$ of the vertices of a graph $G$ with integer colours so that for any path $u_0,\ldots,u_p$ of length at most $\ell$, $\varphi(u_0)\neq\varphi(u_p)$ or $\varphi(u_0)<\max\{\varphi(u_0),\ldots,\varphi(u_p)\}$. We show that, for any fixed integer $\ell\ge 2$, every $n$-vertex planar graph has an $\ell$-ranking using $O(\log n/\log\log\log n)$ colours and this is tight even when $\ell=2$; for infinitely many values of $n$, there… 

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