Rainbow Trees in Hypercubes

TL;DR

This paper proves that in the n-dimensional hypercube, any proper edge-coloring guarantees the existence of rainbow copies of all trees with up to n edges, establishing an optimal result.

Contribution

It establishes the exact conditions under which rainbow trees appear in hypercube edge-colorings, extending understanding of rainbow subgraphs in high-dimensional graphs.

Findings

Every proper edge-coloring of Q_n contains a rainbow copy of every tree with at most n edges.

The result is optimal; Q_n can be colored with n colors avoiding rainbow cycles.

The proof confirms the tightness of the conditions for rainbow trees in hypercubes.

Abstract

We prove that every proper edge-coloring of the $n$-dimensional hypercube $Q_n$ contains a rainbow copy of every tree $T$ on at most $n$ edges. This result is best possible, as $Q_n$ can be properly edge-colored using only $n$ colors while avoiding rainbow cycles.