Coloring the Cube with Rainbow Cycles (original) (raw)

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Keywords: cube, graph coloring

Abstract

For every even positive integer kge4k\ge 4kge4 let f(n,k)f(n,k)f(n,k) denote the minimim number of colors required to color the edges of the nnn-dimensional cube QnQ_nQn, so that the edges of every copy of the kkk-cycle CkC_kCk receive kkk distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that f(n,4)=nf(n,4)=nf(n,4)=n for n=4n=4n=4 or n>5n>5n>5. We consider larger kkk and prove that if kequiv0k \equiv 0kequiv0 (mod 4), then there are positive constants c1,c2c_1, c_2c_1,c_2 depending only on kkk such that$$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For kequiv2k \equiv 2kequiv2 (mod 4), the situation seems more complicated. For the smallest case k=6k=6k=6 we show that 3n-2 \le f(n, 6) < n^{1+o(1)}$$ with the lower bound holding for nge3n \ge 3nge3. The upper bound is obtained from Behrend's construction of a subset of integers with no three term arithmetic progression.

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How to Cite

Mubayi, D., & Stading, R. (2013). Coloring the Cube with Rainbow Cycles. The Electronic Journal of Combinatorics, 20(2), #P4. https://doi.org/10.37236/2957