A Remark on Rainbow 6-Cycles in Hypercubes

Abstract

We call an edge-coloring of a graph [Formula: see text] a rainbow coloring if the edges of [Formula: see text] are colored with distinct colors. For every even positive integer [Formula: see text], let [Formula: see text] denote the minimum number of colors required to color the edges of the [Formula: see text]-dimensional cube [Formula: see text], so that every copy of [Formula: see text] is rainbow. Faudree et al. [6] proved that [Formula: see text] for [Formula: see text] or [Formula: see text]. Mubayi et al. [8] showed that [Formula: see text]. In this note, we show that [Formula: see text]. Moreover, we obtain the number of 6-cycles of [Formula: see text].