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.

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 $n$.

Equations

$$\big{\{}i\in\{0,...,n-1\}:x(i)\neq y(i)\big{\}}.$$

Full text

1. The Hamming Graph $H(n,k)$

We construct graph on the vertex set $\{0,1\}^{n}$ where $n$ is a positive integer. For $x,y\in\{0,1\}^{n}$ the Hamming distance of $x,y$ is the cardinality of the set

[TABLE]

That is, we count the positions on which $x$ and $y$ do not agree.

Fix a positive integer $k\leq n$. Two distinct elements of $\{0,1\}^{n}$ form an edge if their Hamming distance is at most $k$ (so they are in some sense “close” to each other). We denote the resulting graph on $\{0,1\}^{n}$ by $H(n,k)$.

We say that a finite graph $G=(V,E)$ is Hamming-representable if there are positive integers $k\leq n$ such that $G$ is isomorphic to an induced subgraph of $H(n,k)$.

As an easy example, we show that the following $3$-point graph can be embedded into $H(2,1)$:

abc

The solution is best shown in the following picture, where it is easily seen that points connected with an edge have Hamming distance $1$ and points not connected have Hamming distance $2$:

000110

As a further example, note that $H(n,n)$ is isomorphic to $K_{2^{n}}$, the complete graph on $2^{n}$ vertices.

Some notation: By $\mathrm{Mat}(\{0,1\},n\times m)$ we denote the set of $n\times m$-matrices with entries in $\{0,1\}$. We identify $\mathrm{Mat}(\{0,1\},n\times m)$ with $\{0,1\}^{nm}$ via the canonical bijection.

2. The Result

Proposition 2.1**.**

Every finite graph $G=(V,E)$ is Hamming-representable.

Proof. We embed $G$ into $H(|E|\cdot(|V|-1),\ 2|E|-2)$. To each vertex $v$ of $G$, we will associate an $|E|\times(|V|-1)$ matrix $M_{v}$ with rows indexed by the edges of $G$. There will be a single $1$ in each row, with all other entries in that row equal to [math].

If $v\in e$, then the $1$ in row $e$ of $M_{v}$ will be in the first column. If not, we will place a $1$ in one of the other $|V|-2$ columns, so that each of the non-endpoints of $e$ gets a $1$ in a different position of row $e$.

If $v$ and $w$ are not joined by an edge, the Hamming distance between $M_{v}$ and $M_{w}$ is $2|E|$ because they have no $1$’s in common; if they are joined, then the Hamming distance is $2|E|-2$. $\Box$.

3. Possible use cases

Representing graphs as subgraphs of some $H(n,k)$ can be useful in applications in computer science: the Hamming distance is computed by bitwise XOR, the fastest operation a CPU can do. So given two vertices represented by $n$-bit strings, it can be very quickly determined whether they form an edge (i.e. whether their Hamming distance is smaller than the limit given in $k$).

Moreover, for some graphs $G=(V,E)$ with $|V|=n$ we can represent the graph using bit strings of length ${\mathcal{O}}(\log n)$, making this technique potentially interesting for memory management.

4. Open questions

We define the Hamming dimension of a graph $G=(V,E)$ to be the minimum positive integer $n$ such that there is $k\leq n$ such that $G$ can be embedded into some induced subgraph of $H(n,k)$, and denote this by $\mathrm{dim}(G)$ Questions:

1. (1)

If $G=(V,E)$ is a graph with $n=|V|$, do we necessarily have $\mathrm{dim}(G)\leq n$? If not, can we at least acheive for $\mathrm{dim}(G)$ to be ${\mathcal{O}}(|E|\log|V|)$? 2. (2)

Given graphs $G,H$ what is $\mathrm{dim}(G\times H)$ in terms of $\mathrm{dim}(G),\mathrm{dim}(H)$, where $G\times H$ denotes the categorical product? 3. (3)

How (if at all) does $\mathrm{dim}(G)$ relate to the chromatic number $\chi(G)$?

5. Acknowledgement

I am grateful to Prof. David Speyer of the University of Michigan, Ann Arbor, USA, for the construction used in the proof of Proposition 2.1.