Rainbow Trees in Hypercubes

2025-08-19

AI Summary

This paper investigates the existence of rainbow trees in properly edge-colored $n$-dimensional hypercubes $Q_n$. **Problem:** We ask for the maximum number of edges in a tree $T$ that is guaranteed to admit a rainbow embedding into any proper edge coloring of $Q_n$. **Method:** Leveraging the recursive structure of $Q_n$, a tree-embedding lemma, and extremal combinatorial analysis, we provide a constructive proof. **Contribution/Results:** We establish that every properly edge-colored $Q_n$ contains a rainbow copy of any tree $T$ with at most $n$ edges; this bound is tight—there exist proper colorings of $Q_n$ admitting no rainbow embedding of any $(n+1)$-edge tree, and even rainbow cycles cannot be guaranteed. This yields the first exact threshold—the edge count $n$—for rainbow tree embeddability in $Q_n$, resolving a long-standing optimality question in rainbow subgraph theory and extending its applicability to high-dimensional regular graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Finding rainbow trees in hypercube edge-colorings
• Proving optimal tree containment in hypercubes
• Establishing tight bounds for rainbow subgraphs

Innovation

• Hypercube edge-coloring technique
• Rainbow tree embedding method
• Optimal color avoidance strategy