Improved bound for Hadwiger's conjecture

Abstract

Hadwiger conjectured in 1943 that for every integer $t\ge 1$, every graph with no $K_t$ minor is $(t-1)$-colorable. Kostochka, and independently Thomason, proved every graph with no $K_t$ minor is $O(t(\log t{)}^{1/2})$-colorable. Recently, Postle improved it to $O(t(\log \log t{)}^6)$-colorable. In this paper, we show that every graph with no $K_t$ minor is $O(t(\log \log t{)}^5)$-colorable.