Breaking the degeneracy barrier for coloring graphs with no $K_t$ minor

TL;DR

This paper improves the upper bound on the chromatic number of graphs with no $K_t$ minor, showing they are colorable with fewer colors than previously known, specifically $O(t( ext{log } t)^{eta})$ for $eta > 1/4$.

Contribution

It provides the first improvement over the Kostochka-Thomason bound for graphs with no $K_t$ minor, reducing the order of magnitude of the coloring bound.

Findings

Graphs with no $K_t$ minor are $O(t( ext{log } t)^{eta})$-colorable for $eta > 1/4$

Improves the known bound from $O(t ext{sqrt(log } t))$

Establishes a new asymptotic upper bound on the chromatic number

Abstract

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\geq 1$. 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. We show 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 Kostochka-Thomason bound.