Abstract. 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 χ(G k ) unbounded and χ (G k ) ≥ cχ(G k ) log χ(G k ). We also provide an upper bound, χ (G k ) < χ(G k ) 3 for k > 1.
We consider the topology of real no k-equal spaces via the theory of cellular spanning trees. Our main theorem proves that the rank of the (k − 2)-dimensional homology of the no k-equal subspace of R is equal to the number of facets in a k-dimensional spanning tree of the k-skeleton of the n-dimensional hypercube.
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