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. More recently, the second author showed they are $O(t (\log \log t)^{6})$-colorable. Our main technical result is 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, 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. Second, 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. 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 any fixed $r$ and sufficiently large $t$; more specifically, there exists $C\ge 1$ such that for every $r\ge 1$, there exists $t_r$ such that for all $t\ge t_r$, every $K_r$-free $K_t$-minor-free graph is $Ct$-colorable. 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 $O(d \log^3 t)$ vertices with average degree $\Omega(d)$.

中文翻译：

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减少线性 Hadwiger 猜想对小图进行着色

1943 年，Hadwiger 推测每个没有 $K_t$ 小调的图对于每个 $t\ge 1$ 都是 $(t-1)$-可着色的。在 1980 年代，Kostochka 和 Thomason 独立证明了每个没有 $K_t$ 次要的图都有平均度 $O(t\sqrt{\log t})$，因此是 $O(t\sqrt{\log t})$ -可着色。最近，Norin、Song 和第二作者表明，对于每个 $\beta > 1/4$，每个没有 $K_t$ 次要的图都是 $O(t(\log t)^{\beta})$-colorable，使得$O(t\sqrt{\log t})$ 边界数量级上的第一个改进。最近，第二作者表明它们是 $O(t (\log \log t)^{6})$-colorable。我们的主要技术结果是 $K_t$-minor-free 图的色数有界 $O(t(1+f(G,t))$ 其中 $f(G, t)$ 是 $\frac{\chi(H)}{a}$ 在所有 $a\ge \frac{t}{\sqrt{\log t}}$ 和 $K_a$-minor-free 中的最大值$G$ 的小子图 $H$（即 $O(a\log^4 a)$ 顶点）。这有许多有趣的推论。首先，它表明证明线性哈德维格猜想（即 $K_t$-minor-free 图是 $O(t)$-colorable）简化为证明小图。其次，使用当前最著名的对无 $K_t$-minor-free 图着色的边界，我们证明 $K_t$-minor-free 图是 $O(t\log\log t)$-colorable。第三，我们证明了群数最多为 $\sqrt{\log t}/ (\log \log t)^2$ 的 $K_t$-minor-free 图是 $O(t)$-可着色的。这意味着我们的最后一个推论，即线性哈维格猜想对于任何固定的 $r$ 和足够大的 $t$ 的无 $K_r$ 的图成立；进一步来说，存在 $C\ge 1$ 使得对于每个 $r\ge 1$，存在 $t_r$ 使得对于所有 $t\ge t_r$，每个 $K_r$-free $K_t$-minor-free 图是$Ct$-可着色。证明主定理的一个关键是一个新的独立结果，即每个平均度数为 $d=\Omega(t)$ 的 $K_t$-minor-free 图在 $O(d \log^3 t)$ 顶点上都有一个子图平均度数为 $\Omega(d)$。