Improved lower bound for the list chromatic number of graphs with no K t minor
Raphael Steiner · Combinatorics Probability Computing · 2022
Abstract Hadwiger’s conjecture asserts that every graph without a $K_t$ -minor is $(t-1)$ -colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$ -minor has list chromatic number at most $ct$ . More specifically, they also conjectured that this holds for $c=\frac{3}{2}$ . Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$ -minor is at least $(2-o(1))t$ , and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$ . Our lower-bound examples are obtained via the probabilistic method.