Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs
Michelle Delcourt, Luke Postle
TL;DR
This work addresses Hadwiger's conjecture by proving a tight, scalable coloring bound for $K_t$-minor-free graphs. The authors introduce a main technical bound $\nchi(G) \le C t (1+f(G,t))$, where $f(G,t)$ captures how well small $K_a$-minor-free subgraphs can be colored relative to their size, and they show $\nchi(G) = O(t\log\log t)$ by combining new subgraph coloring bounds with a density-to-minor framework. A central contribution is a small-dense-subgraph theorem and a robust connectivity toolkit that enable constructing and linking minors in novel ways, including sequential and recursive strategies that avoid prior bottlenecks. The paper derives notable corollaries: Linear Hadwiger holds if proving it for small graphs suffices, Linear Hadwiger holds for every fixed $K_r$-free class, and the sublogarithmic clique bound interacts favorably with existing density results to yield stronger colorability conclusions for certain graph families.
Abstract
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 $O(t(\log t)^β)$-colorable for every $β> 1/4$, 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 $\frac{χ(H)}{a}$ 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 $d=Ω(t)$ has a subgraph on $O(t \log^3 t)$ vertices with average degree $Ω(d)$.