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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
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References:
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