Coloring the cube with rainbow cycles

TL;DR

This paper investigates the minimum number of colors needed to edge-color high-dimensional cubes so that all cycles of a given length have edges of distinct colors, providing bounds for different cycle lengths and using combinatorial constructions.

Contribution

It establishes new bounds on the coloring function for even cycle lengths in hypercubes, extending previous results and applying advanced combinatorial set constructions.

Findings

For k divisible by 4, f(n,k) is on the order of n^{k/4}.

For k ≡ 2 mod 4, f(n,k) is between n and n^{1+o(1)}.

Uses Bose-Chowla and Behrend constructions for bounds.

Abstract

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gy\'arf\'as, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ 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 $k \equiv 2$ (mod 4), the situation seems more complicated. For the smallest case k=6 we show that $$n \le f(n, 6) < n^{1+o(1)}.$$ The upper bound is obtained from Behrend's construction of a subset of the integers with no three term arithmetic progression.