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A Remark on Rainbow 6-Cycles in Hypercubes

    We call an edge-coloring of a graph G a rainbow coloring if the edges of G are colored with distinct colors. For every even positive integer k4, let f(n,k) denote the minimum number of colors required to color the edges of the n-dimensional cube Qn, so that every copy of Ck is rainbow. Faudree et al. [6] proved that f(n,4)=n for n=4 or n>5. Mubayi et al. [8] showed that nf(n,6)<n1+o(1). In this note, we show that f(n,6)2n1. Moreover, we obtain the number of 6-cycles of Qn.

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