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

Graph Theory Seminar
Tuesday, September 15, 2020 - 3:45pm for 1 hour (actually 50 minutes)
Location For password, please email Anton Bernshteyn (bahtoh ~at~
Zi-Xia Song – University of Central Florida – Zixia.Song@ucf.edu
Anton Bernshteyn

Hadwiger's conjecture from 1943 states that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the early 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.  In this talk, 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 is joint work with  Sergey Norin and Luke Postle.