• Corpus ID: 219965889

Further progress towards Hadwiger's conjecture

@article{Postle2020FurtherPT,
  title={Further progress towards Hadwiger's conjecture},
  author={Luke Postle},
  journal={ArXiv},
  year={2020},
  volume={abs/2006.11798}
}
  • L. Postle
  • Published 21 June 2020
  • Computer Science, Mathematics
  • ArXiv
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt… 

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  • Computer Science, Mathematics
    ArXiv
  • 2020
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  • Mathematics
    Combinatorics, Probability and Computing
  • 2022
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Improving on the second part of their argument, Norin and Song prove that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1}{4}$.

An even better Density Increment Theorem and its application to Hadwiger's Conjecture

  • L. Postle
  • Computer Science, Mathematics
    ArXiv
  • 2020
Every graph with no $K_t$ minor is $O(t (\log \log t)^{6})$-colorable, making the first improvement on the order of magnitude of the O(t\sqrt{\log t})$ bound.

Breaking the degeneracy barrier for coloring graphs with no $K_t$ minor

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$

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This is a survey of Hadwiger’s conjecture from 1943, that for all t ≥ 0, every graph either can be t-coloured, or has a subgraph that can be contracted to the complete graph on t + 1 vertices. This

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In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough'' structure of graphs excluding a fixed minor. This result was used to prove