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：对任意$t\ge 1$，不含$K_t$子式的图均为$(t-1)$-可着色的。20世纪80年代，Kostochka和Thomason独立证明：不含$K_t$子式的图平均度均为$O(t\sqrt{\log t})$，因而是$O(t\sqrt{\log t})$-可着色的。近期，Norin、Song及第二作者证明了：对任意$\beta > 1/4$，不含$K_t$子式的图均为$O(t(\log t)^{\beta})$-可着色的，首次改进了$O(t\sqrt{\log t})$界值的量级。本文首个主要结果是：不含$K_t$子式的图均为$O(t\log\log t)$-可着色的。这源于我们的核心技术结论：不含$K_t$子式图的色数被$O(t(1+f(G,t)))$所界定，其中$f(G,t)$定义为所有$a\ge \frac{t}{\sqrt{\log t}}$及$G$中不含$K_a$子式的小规模子图$H$（即顶点数$O(a\log^4 a)$）上$\frac{\chi(H)}{a}$的最大值。该结论具有多个有趣推论。首先，如前述，利用当前已知的最佳小规模$K_t$-子式自由图着色界，我们证明$K_t$-子式自由图为$O(t\log\log t)$-可着色。其次，它表明线性哈德维格猜想（即$K_t$-子式自由图为$O(t)$-可着色）可归约为对小图证明该猜想。第三，我们证明团数不超过$\sqrt{\log t}/(\log\log t)^2$的$K_t$-子式自由图为$O(t)$-可着色，进而推出最终推论：对任意固定$r$，线性哈德维格猜想在$K_r$-自由图中成立。证明主定理的关键在于一个新的独立结果：平均度$d=\Omega(t)$的任意$K_t$-子式自由图，都存在一个顶点数$O(t\log^3 t)$且平均度$\Omega(d)$的子图。