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arXiv · 2308.15721

Clustered Colouring of Odd-$H$-Minor-Free Graphs

Robert Hickingbotham·Dong Yeap Kang·Sang-il Oum·Raphael Steiner·David R. Wood
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Abstract

The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.

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Robert Hickingbotham, Dong Yeap Kang, Sang-il Oum, Raphael Steiner, David R. Wood. 2023-08-30. Clustered Colouring of Odd-$H$-Minor-Free Graphs. https://arxiv.org/abs/2308.15721

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