Further progress towards Hadwiger's conjecture
Abstract
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t1)$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 $O(t\sqrt{\log t})$ bound. Building on that work, we show in this paper that every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$colorable for every $\beta > 0$. More specifically in conjunction with another paper by the author, they are $O(t \cdot (\log \log t)^{18})$colorable.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.11798
 Bibcode:
 2020arXiv200611798P
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics
 EPrint:
 Merged into arXiv:2108.01633