A note on list-coloring powers of graphs

Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some k such that all kth power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant c such that for any k there is a family of graphs G with χ(Gk) unbounded and χℓ(Gk)≥cχ(Gk)logχ(Gk). We also provide an upper bound, χℓ(Gk)<χ(Gk)3 for k>1.