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Clustered Colouring of Odd-$H$-Minor-Free Graphs
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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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3-Colouring Planar Graphs
Every n-vertex planar graph can be 3-coloured so that each monochromatic connected component has at most O(n^{4/9}) vertices, improving the previous O(n^{1/2}) bound.
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