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 $χ_{\ell}(G^k)\geq c χ(G^k) \log χ(G^k)$. We also provide an upper bound, $χ_{\ell}(G^k)<χ(G^k)^3$ for $k>1$.

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