Book ; Online: Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs
2021
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 ...
| 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 $O(t(\log t)^{\beta})$-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 $\Omega(d)$. Comment: 25 pages. Updated in response to referee comments. This and the two previous versions add the necessary results from arXiv:2006.11798 in order to create a self-contained standalone paper. arXiv admin note: text overlap with arXiv:2006.11798, arXiv:2010.05999 |
|---|---|
| Keywords | Mathematics - Combinatorics ; Computer Science - Discrete Mathematics ; 05C15 ; 05C83 |
| Subject code | 511 |
| Publishing date | 2021-08-03 |
| Publishing country | us |
| Document type | Book ; Online |
| Database | BASE - Bielefeld Academic Search Engine (life sciences selection) |
Full text online
More links
Kategorien
Inter-library loan at ZB MED
Your chosen title can be delivered directly to ZB MED Cologne location if you are registered as a user at ZB MED Cologne.