Disproof of a conjecture by Woodall on the choosability of \(K_{s,t}\)-minor-free graphs (Q2153396)

Summary: \textit{D. R. Woodall} [Lond. Math. Soc. Lect. Note Ser. 288, 269--301 (2001; Zbl 0979.05047)], in a survey article about list coloring, conjectured that for every pair of integers \(s, t \geqslant 1\), all graphs without a \(K_{s,t}\)-minor are \((s+t-1)\)-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants \(\varepsilon>0\) and \(C \geqslant 1\) there exists \(N=N (\varepsilon, C) \in \mathbb{N}\) such that for all integers \(s\), \(t\) with \(N \leqslant s \leqslant t \leqslant Cs\) there exists a graph without a \(K_{s,t}\)-minor and list chromatic number greater than \((1-\varepsilon)(2s+t)\).