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References (16)
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Question posed at the BIRS meeting on Graph Colouring and Structure -
S. Norin, Zi-Xia Song (2019)
Breaking the degeneracy barrier for coloring graphs with no K minorAdvances in Mathematics
- Publisher
- Wiley
- Copyright
- © 2022 London Mathematical Society.
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms.12569
- Publisher site
- See Article on Publisher Site
Abstract
Resolving a problem raised by Norin in 2020, we show that for each k∈N$k \in \mathbb {N}$, the minimal f(k)∈N$f(k) \in \mathbb {N}$ with the property that every graph G$G$ with chromatic number at least f(k)+1$f(k)+1$ contains a subgraph H$H$ with both connectivity and chromatic number at least k$k$ satisfies f(k)⩽7k$f(k) \leqslant 7k$. This result is best‐possible up to multiplicative constants, and sharpens earlier results of Alon–Kleitman–Thomassen–Saks–Seymour from 1987 showing that f(k)=O(k3)$f(k) = O(k^3)$, and of Chudnovsky–Penev–Scott–Trotignon from 2013 showing that f(k)=O(k2)$f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we additionally show that for each k∈N$k \in \mathbb {N}$, the minimal fℓ(k)∈N$f_\ell (k) \in \mathbb {N}$ with the property that every graph G$G$ with list chromatic number at least fℓ(k)+1$f_\ell (k)+1$ contains a subgraph H$H$ with both connectivity and list chromatic number at least k$k$ is well‐defined and satisfies fℓ(k)⩽4k$f_\ell (k) \leqslant 4k$. This result is again best‐possible up to multiplicative constants; here, unlike with f(·)$f(\cdot )$, even the existence of fℓ(·)$f_\ell (\cdot )$ appears to have been previously unknown.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Jun 1, 2022