In 1943, Hadwiger conjectured that every graph with no $K_{t+1}$ minor is t-colorable for every $t\ge 0$. While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and thus is $O(t\sqrt{\log t})$-list-colorable.

Recently, the authors and Song proved that every graph with no $K_t$ minor is $O(t{(\log t)}^{\beta })$-colorable for every $\beta >\frac{1}{4}$. Here, we build on that result to show that every graph with no $K_t$ minor is $O(t{(\log t)}^{\beta })$-list-colorable for every $\beta >\frac{1}{4}$.

Our main new tool is an upper bound on the number of vertices in highly connected $K_t$-minor-free graphs: We prove that for every $\beta >\frac{1}{4}$, every $\mathrm{\Omega }(t{(\log t)}^{\beta })$-connected graph with no $K_t$ minor has $O(t{(\log t)}^{7/4})$ vertices.

1943年，哈德维格（Hadwiger）猜想每个图都没有 ${ķ}_{Ť+\mathrm{1个}}$未成年人是牛逼，每-colorable$Ť\ge 0$。虽然哈德维格的猜想不能满足于列表着色，但线性减弱被认为是正确的。在1980年代，Kostochka和Thomason独立证明了每张图都没有${ķ}_{Ť}$ 未成年人平均学位 $Ø（Ť\sqrt{\mathrm{日志}Ť}）$ 因此是 $Ø（Ť\sqrt{\mathrm{日志}Ť}）$-list-colorable。

最近，作者和宋证明了每个图都没有 ${ķ}_{Ť}$ 未成年人是 $Ø（Ť{（\mathrm{日志}Ť）}^{\beta }）$-每个颜色 $\beta >\frac{\mathrm{1个}}{4}$。在此，我们以该结果为基础来显示每个图都没有${ķ}_{Ť}$ 未成年人是 $Ø（Ť{（\mathrm{日志}Ť）}^{\beta }）$-list-colorable每个 $\beta >\frac{\mathrm{1个}}{4}$。

我们的主要新工具是高度连接的顶点数量的上限 ${ķ}_{Ť}$-次要图表：我们证明了 $\beta >\frac{\mathrm{1个}}{4}$， 每一个 $\mathrm{\Omega }（Ť{（\mathrm{日志}Ť）}^{\beta }）$连接图无 ${ķ}_{Ť}$ 未成年人 $Ø（Ť{（\mathrm{日志}Ť）}^{7/4}）$ 顶点。