A note on sequences variant of irregularity strength for hypercubes

TL;DR

This paper investigates edge colorings of hypercubes to distinguish vertices by their palette sequences, proving that two colors suffice generally, and that n colors suffice for proper colorings when n ≥ 5.

Contribution

It establishes minimal color requirements for vertex distinction in hypercubes via palette sequences, including both general and proper edge coloring cases.

Findings

Two colors are enough to distinguish all vertices by palettes in any hypercube.

For proper edge colorings of hypercubes with n ≥ 5, n colors suffice for vertex distinction.

The results extend understanding of irregularity strength variants for hypercubes.

Abstract

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