  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 $O(t\sqrt{\log t})$ bound.
Building on that work, we previously showed that every graph with no $K_t$
minor is $O(t (\log t)^{\beta})$-colorable for every $\beta > 0$. More
specifically, they are $O(t \cdot (\log \log t)^{6})$-colorable. In this paper,
we extend that work to the list and odd generalizations of Hadwiger's
conjecture.
