# Improved bounds for centered colorings

@inproceedings{Dkebski2019ImprovedBF, title={Improved bounds for centered colorings}, author={Michal Dkebski and Stefan Felsner and Piotr Micek and Felix Schroder}, booktitle={ACM-SIAM Symposium on Discrete Algorithms}, year={2019} }

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p…

## 29 Citations

### Two lower bounds for $p$-centered colorings

- 2020

Mathematics

Discret. Math. Theor. Comput. Sci.

There are graphs of maximum degree $\Delta$ that require $\Omega(\Delta^{2-1/p} p \ln^{- 1/p}\Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.

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

- 2019

Mathematics

SODA

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.

### Improved Bounds for Weak Coloring Numbers

- 2022

Mathematics

Electron. J. Comb.

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…

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

- 2019

Mathematics

For p ∈ N, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once…

### A Fast Algorithm for the Product Structure of Planar Graphs

- 2021

Computer Science, Mathematics

Algorithmica

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) .

### Asymptotically Optimal Vertex Ranking of Planar Graphs

- 2020

Mathematics, Computer Science

ArXiv

New sublogarithmic upper bounds on the number of colours needed for $\ell$-rankings of apex minor-free graphs and $k$-planar graphs are obtained and these upper bounds are constructive and give $O(n\log n)$-time algorithms.

### The structure of k-planar graphs

- 2019

Mathematics

ArXiv

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.

### Improved product structure for graphs on surfaces

- 2022

Mathematics

Discret. Math. Theor. Comput. Sci.

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…

### Graph product structure for non-minor-closed classes

- 2023

Mathematics

Journal of Combinatorial Theory, Series B

### Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

- 2022

Mathematics

J. Graph Theory

We show that for any fixed integer m ≥ 1 $m\ge 1$ , a graph of maximum defiggree Δ ${\rm{\Delta }}$ has a coloring with O ( Δ ( m + 1 ) ∕ m ) $O({{\rm{\Delta }}}^{(m+1)\unicode{x02215}m})$ colors in…

## 32 References

### Two lower bounds for $p$-centered colorings

- 2020

Mathematics

Discret. Math. Theor. Comput. Sci.

There are graphs of maximum degree $\Delta$ that require $\Omega(\Delta^{2-1/p} p \ln^{- 1/p}\Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.

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

- 2019

Mathematics

SODA

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.

### Improved Bounds for Weak Coloring Numbers

- 2022

Mathematics

Electron. J. Comb.

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…

### A Fast Algorithm for the Product Structure of Planar Graphs

- 2021

Computer Science, Mathematics

Algorithmica

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) .

### Polynomial Treedepth Bounds in Linear Colorings

- 2021

Mathematics

Algorithmica

P - linear colorings are introduced as an alternative to the commonly used p -centered colorings and a co-NP-completeness reduction for recognizing p -linear colorings is given and discussed.

### Star coloring of graphs

- 2004

Mathematics

J. Graph Theory

The exact value of the star chromatic number of different families of graphs, such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids, is given.

### Layout of Graphs with Bounded Tree-Width

- 2005

Mathematics

SIAM J. Comput.

It is proved that the queue-number is bounded by the tree-width, thus resolving an open problem due to Ganley and Heath and disproving a conjecture of Pemmaraju.

### Planar Graphs have Bounded Queue-Number

- 2019

Mathematics

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

It is proved that every proper minor-closed class of graphs has bounded queue-number, and it is shown that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth.

### Structure theorem and isomorphism test for graphs with excluded topological subgraphs

- 2012

Mathematics

STOC '12

It is proved that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices.