Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs

In 1943, Hadwiger conjectured that every graph with no $Kt$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $Kt$ 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 $Kt$ 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 $Kt$ minor is $O(t\log\log t)$-colorable. This is a corollary of our main technical result that the chromatic number of a $Kt$-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 $Ka$-minor-free subgraphs $H$ of $G$ that are small (i.e. $O(a\log⁴ a)$ vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small $Kt$-minor-free graphs, we show that $Kt$-minor-free graphs are $O(t\log\log t)$-colorable. Second, it shows that proving Linear Hadwiger's Conjecture (that $Kt$-minor-free graphs are $O(t)$-colorable) reduces to proving it for small graphs. Third, we prove that $Kt$-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 $Kr$-free graphs for every fixed $r$. One key to proving the main theorem is a new standalone result that every $Kt$-minor-free graph of average degree $d=\Omega(t)$ has a subgraph on $O(t \log³ t)$ vertices with average degree $\Omega(d)$.