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 k th 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 ( G k ) unbounded and χ ℓ ( G k ) ≥ c χ ( G k ) log χ ( G k ) . We also provide an upper bound, χ ℓ ( G k ) < χ ( G k ) 3 for k > 1 .