# Clustered Graph Coloring and Layered Treewidth

@article{Liu2019ClusteredGC, title={Clustered Graph Coloring and Layered Treewidth}, author={Chun-Hung Liu and David R. Wood}, journal={arXiv: Combinatorics}, year={2019} }

A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies clustered coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption than previous works, where typically the number of colors depends on an excluded minor. This paper focuses on graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with…

## 16 Citations

### Clustered Coloring of Graphs Excluding a Subgraph and a Minor

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A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This…

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The main theorem says that every graph with layered treewidth at most k and with maximum degree at most ∆ is 3 -colorable with clustering O ( k 19 ∆ 37 ) .

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### The cases for t ≤ 3 of Conjecture 1 . 1 follow from the correctness of Hajós ’ conjecture for t ≤ 3

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Hadwiger and Hajós conjectured that for every positive integer t, Kt+1-minor free graphs and Kt+1-topological minor free graphs are properly t-colorable, respectively. Clustered coloring version of…

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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.

### Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

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It is proved that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups.

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