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A note on list-coloring powers of graphs. (English) Zbl 1298.05123
Summary: Recently, S.-J. Kim and B. Park [“Counterexamples to the list square coloring conjecture”, J. Graph Theory (to appear; doi:10.1002/jgt.21802)] 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 $$\chi(G^k)$$ unbounded and $$\chi_\ell(G^k) \geq c \chi(G^k) \log \chi(G^k)$$. We also provide an upper bound, $$\chi_\ell(G^k) < \chi(G^k)^3$$ for $$k > 1$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
choosability; list coloring; chromatic-choosable; graph powers
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##### References:
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