• 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}
}
• Published 13 July 2020
• Mathematics, Computer Science
• ArXiv
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…
10 Citations

## Tables from this paper

This report presents a meta-modelling system that automates the very labor-intensive and therefore time-heavy and therefore expensive and expensive process of manually cataloging and cataloging all the components of a smart phone.
• Mathematics
• 2022
. Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly
• Mathematics
• 2022
A squaregraph is a plane graph in which each internal face is a 4-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the
• Mathematics
SWAT
• 2022
The Product Structure Theorem for planar graphs (Dujmović et al. JACM , 67 (4):22) states that any planar graph is contained in the strong product of a planar 3-tree, a path, and a 3-cycle. We give a
• Mathematics
Discret. Math. Theor. Comput. Sci.
• 2022
Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that
• Mathematics
Electron. J. Comb.
• 2022
This work improves the proved result that for every planar graph G there is a graph H with treewidth at most 8 and a path P such that G ⊆ H P by replacing “treewidthat most 8” by “simple treewitzer at most 6”.
• Computer Science
Discret. Math. Theor. Comput. Sci.
• 2022
This paper answers the question in the negative of whether row treewidth is bounded by a function of layered trewidth and proves an analogous result for layered pathwidth and row pathwidth.
• Mathematics
Electron. J. Comb.
• 2022
Weak coloring numbers generalize the notion of degeneracy of a graph. They were introduced by Kierstead & Yang in the context of games on graphs. Recently, several connections have been uncovered
• Mathematics
• 2021
This paper shows that K t -minor-free (and K s,t -minor-free) graphs G are subgraphs of products of a tree-like graph H (of bounded treewidth) and a complete graph K m . Our results include optimal
• Mathematics
• 2019
It is proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path and implies, amongst other results, that $k$-planar graphs have non-repetitive chromatic number upper-bounded by a function of $k$.

## References

SHOWING 1-10 OF 53 REFERENCES

• Mathematics
SIAM 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.
• Mathematics
SODA
• 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.
• Mathematics
Graphs 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.
• Mathematics
SODA
• 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.
• 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) .
• Mathematics
CALDAM
• 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).
• Mathematics
ArXiv
• 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.
• Mathematics
Combinatorics, 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).
• Computer Science, Mathematics
SODA '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.