Improved bound for Hadwiger's conjecture

TL;DR

This paper improves the upper bound on the chromatic number of graphs excluding a complete graph minor, reducing it from previous bounds involving logarithmic factors to a tighter bound with a smaller exponent on the log log term.

Contribution

The authors establish a new bound of O(t (log log t)^5) for the chromatic number of graphs with no K_t minor, refining previous results.

Findings

Improved the upper bound to O(t (log log t)^5)

Reduced the exponent on the log log factor from 6 to 5

Contributed to the understanding of Hadwiger's conjecture bounds

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.