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# Tutte Colloquium - Luke Postle

Friday, September 11, 2020 — 3:30 PM EDT

Title: Further progress towards Hadwiger's conjecture

 Speaker: Luke Postle Affiliation: University of Waterloo Zoom: Please email Emma Watson.

Abstract:

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 I 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. Here we show that every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$-colorable for every $\beta > 0$; more specifically, they are $O(t (\log \log t)^{6})$-colorable.

### July 2021

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