Note: 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 @g(G^k) unbounded and @g"@?(G^k)>[email protected](G^k)[email protected](G^k). We also provide an upper bound, @g"@?(G^k) 1.