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 second 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. The first main result of this paper is that every graph with no $K_t$ minor is $O(t\log\log t)$-colorable. This is a corollary of our main technical result that the chromatic number of a $K_t$-minor-free graph is bounded by $O(t(1+f(G,t)))$ where $f(G,t)$ is the maximum of $\frac{\chi(H)}{a}$ over all $a\ge \frac{t}{\sqrt{\log t}}$ and $K_a$-minor-free subgraphs $H$ of $G$ that are small (i.e. $O(a\log^4 a)$ vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small $K_t$-minor-free graphs, we show that $K_t$-minor-free graphs are $O(t\log\log t)$-colorable. Second, it shows that proving Linear Hadwiger's Conjecture (that $K_t$-minor-free graphs are $O(t)$-colorable) reduces to proving it for small graphs. Third, we prove that $K_t$-minor-free graphs with clique number at most $\sqrt{\log t}/ (\log \log t)^2$ are $O(t)$-colorable. This implies our final corollary that Linear Hadwiger's Conjecture holds for $K_r$-free graphs for every fixed $r$. One key to proving the main theorem is a new standalone result that every $K_t$-minor-free graph of average degree $d=\Omega(t)$ has a subgraph on $O(t \log^3 t)$ vertices with average degree $\Omega(d)$.

翻译：1943年，哈德维格猜想：对于每个 $t\ge 1$，不含 $K_t$ minors 的图都是 $(t-1)$-可着色的。20世纪80年代，Kostochka 与 Thomason 独立证明：每个不含 $K_t$ minors 的图平均度数至多为 $O(t\sqrt{\log t})$，因此是 $O(t\sqrt{\log t})$-可着色的。近期，Norin、Song 与第二作者证明：对任意 $\beta > 1/4$，不含 $K_t$ minors 的图是 $O(t(\log t)^{\beta})$-可着色的，这是对 $O(t\sqrt{\log t})$ 阶上界的首次改进。本文第一个主要结果是：每个不含 $K_t$ minors 的图都是 $O(t\log\log t)$-可着色的。这是如下关键技术结果的推论：$K_t$-minor-free 图的色数不超过 $O(t(1+f(G,t)))$，其中 $f(G,t)$ 定义为对所有 $a\ge \frac{t}{\sqrt{\log t}}$ 及 $G$ 中满足顶点规模 $O(a\log^4 a)$ 的 $K_a$-minor-free 子图 $H$，取 $\frac{\chi(H)}{a}$ 的最大值。该结果蕴含若干有趣推论。首先，如前所述，利用当前对小型 $K_t$-minor-free 图的最优染色上界，我们证明 $K_t$-minor-free 图是 $O(t\log\log t)$-可着色的。其次，它表明证明线性哈德维格猜想（即 $K_t$-minor-free 图是 $O(t)$-可着色的）可简化为对小图的证明。第三，我们证明团数不超过 $\sqrt{\log t}/(\log\log t)^2$ 的 $K_t$-minor-free 图是 $O(t)$-可着色的。由此得到最终推论：对任意固定 $r$，线性哈德维格猜想对 $K_r$-free 图成立。证明主要定理的关键在于一个独立的新结果：每个平均度数 $d=\Omega(t)$ 的 $K_t$-minor-free 图，都存在一个顶点规模为 $O(t \log^3 t)$ 且平均度数为 $\Omega(d)$ 的子图。