• Corpus ID: 220496342

Asymptotically Optimal Vertex Ranking of Planar Graphs

@article{Bose2020AsymptoticallyOV,
  title={Asymptotically Optimal Vertex Ranking of Planar Graphs},
  author={Prosenjit Bose and Vida Dujmovi'c and Mehrnoosh Javarsineh and Pat Morin},
  journal={ArXiv},
  year={2020},
  volume={abs/2007.06455}
}
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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References

SHOWING 1-10 OF 53 REFERENCES

Rankings of Graphs

It is shown that the vertex ranking number $\chi_{r}(G)$ can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k.

Polynomial bounds for centered colorings on proper minor-closed graph classes

The first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes are provided, which answers an open problem posed by Dvoř{a}k.

Graph 2-rankings

The 2-ranking number of G, denotedchi _{2}(G)$$χ2(G), is the minimum number of parts needed for a 2- ranking and the existence of a graph G with maximum degree k is shown.

Improved bounds for centered colorings

Upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes are given.

A Fast Algorithm for the Product Structure of Planar Graphs

  • Pat Morin
  • Computer Science, Mathematics
    Algorithmica
  • 2021
The proof given by Dujmović et al. is based on a similar decomposition of Pilipczuk and Siebertz (SODA2019) which is constructive and leads to an O ( n 2) time algorithm for finding H and the mapping from V ( G ) onto V ( H ⊠ P) .

Complexity of Restricted Variant of Star Colouring

It is shown that for k, it is NP-complete to decide whether a given planar bipartite graph of maximum degree k and girth at least six is 3-rs colourable, and thereby answer the problem posed by Shalu and Sandhya (Graphs and Combinatorics 2016).

The structure of k-planar graphs

It is proved that every planar graph is a subgraph of the strong product of a graph of treewidth $O(k^5)$ and a path, which is the first result of this type for a non-minor-closed class of graphs.

Conflict-Free Colourings of Graphs and Hypergraphs

An efficient deterministic algorithm is given to find a conflict-free colouring of the vertices of a hypergraph H if each hyperedge E of H contains a vertex of ‘unique’ colour that does not get repeated in E, and the smallest number of colours required is denoted by χCF(H).

Optimal edge ranking of trees in polynomial time

This paper gives the first polynomial-time algorithm to find an optimal edge ranking of a tree, placing the problem in P and proves that a natural decision problem emerging from the sequential algorithm is P-complete.

On vertex rankings of graphs and its relatives

...