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

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. 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.

This version adds a new coauthor and significantly strengthens the main result by combining the previous version with arXiv:1911.01491. Several other major changes in content and presentation are made