Further progress towards Hadwiger's conjecture

27 October 2020
15:30
Luke Postle

Further Information: 

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

In 1943, Hadwiger conjectured that every graph with no Kt 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 $O(t(\log t)^{1/2})$ and hence is $O(t(\log t)^{1/2)}$-colorable. Recently, Norin, Song and I 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 $O(t(\log t)^{1/2})$ bound. Here we show that every graph with no $K_t$ minor is $O(t (\log t)^\beta)$-colorable for every $\beta > 0$; more specifically, they are $O(t (\log \log t)^6)$-colorable.

  • Combinatorial Theory Seminar