# 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

- MathematicsSIAM J. Discret. Math.
- 1998

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

- MathematicsSODA
- 2019

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

- MathematicsGraphs Comb.
- 2019

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

- MathematicsSODA
- 2020

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

- Computer Science, MathematicsAlgorithmica
- 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

- MathematicsCALDAM
- 2020

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

- MathematicsArXiv
- 2019

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

- MathematicsCombinatorics, Probability and Computing
- 2009

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

- Computer Science, MathematicsSODA '93
- 1993

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.