A note on sequences variant of irregularity strength for hypercubes

Abstract

Let $f:E\to \{1,2,\dots ,k\}$ be an edge-coloring of the n-dimension hypercube $H_n$. By the palette at a vertex v we mean the sequence $\left(f\right(e_1\left(v\right)),f(e_1\left(v\right)),\dots ,f(e_n\left(v\right)\left)\right)$, where $e_i\left(v\right)$ is the edge incident to v that connects vertices differing in the ith element. In this paper, we show that two colors are enough to distinguish all vertices of the n-dimensional hypercube $H_n$ ($n\ge 2$) by their palettes. We also show that if f is a proper edge-coloring of the hypercube $H_n$ ($n\ge 5$), then n colors suffice to distinguish all vertices by their palettes.