# Graph embeddings into Hamming spaces

## TL;DR

This paper demonstrates that any simple, undirected graph can be embedded into a Hamming space of binary vectors with the Hamming distance, using an injective map, for sufficiently large dimensions.

## Contribution

It introduces a universal embedding of all graphs into Hamming spaces, simplifying graph representation in a discrete metric space.

## Key findings

- Any graph can be embedded into a Hamming space with sufficiently large dimension.
- The embedding is injective, preserving graph structure in the Hamming metric.
- This provides a simple, universal method for graph embedding into binary spaces.

## Abstract

Graph embeddings deal with injective maps from a given simple, undirected graph $G=(V,E)$ into a metric space, such as $\mathbb{R}^n$ with the Euclidean metric. This concept is widely studied in computer science, see \cite{ge1}, but also offers attractive research in pure graph theory \cite{ge2}. In this note we show that any graph can be embedded into a particularly simple metric space: $\{0,1\}^n$ with the Hamming distance, for large enough $n$.