Graph embeddings into Hamming spaces
Dominic van der Zypen

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 into a metric space, such as 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: with the Hamming distance, for large enough .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Computational Geometry and Mesh Generation
Graph embeddings into Hamming spaces
Dominic van der Zypen
Swiss Armed Forces, CH-3003 Bern, Switzerland
Abstract.
Graph embeddings deal with injective maps from a given simple, undirected graph into a metric space, such as with the Euclidean metric. This concept is widely studied in computer science, see [1], but also offers attractive research in pure graph theory [2]. In this note we show that any graph can be embedded into a particularly simple metric space: with the Hamming distance, for large enough .
1. The Hamming Graph
We construct graph on the vertex set where is a positive integer. For the Hamming distance of is the cardinality of the set
[TABLE]
That is, we count the positions on which and do not agree.
Fix a positive integer . Two distinct elements of form an edge if their Hamming distance is at most (so they are in some sense “close” to each other). We denote the resulting graph on by .
We say that a finite graph is Hamming-representable if there are positive integers such that is isomorphic to an induced subgraph of .
As an easy example, we show that the following -point graph can be embedded into :
abc
The solution is best shown in the following picture, where it is easily seen that points connected with an edge have Hamming distance and points not connected have Hamming distance :
000110
As a further example, note that is isomorphic to , the complete graph on vertices.
Some notation: By we denote the set of -matrices with entries in . We identify with via the canonical bijection.
2. The Result
Proposition 2.1**.**
Every finite graph is Hamming-representable.
Proof. We embed into . To each vertex of , we will associate an matrix with rows indexed by the edges of . There will be a single in each row, with all other entries in that row equal to [math].
If , then the in row of will be in the first column. If not, we will place a in one of the other columns, so that each of the non-endpoints of gets a in a different position of row .
If and are not joined by an edge, the Hamming distance between and is because they have no ’s in common; if they are joined, then the Hamming distance is . .
3. Possible use cases
Representing graphs as subgraphs of some 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 -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 ).
Moreover, for some graphs with we can represent the graph using bit strings of length , making this technique potentially interesting for memory management.
4. Open questions
We define the Hamming dimension of a graph to be the minimum positive integer such that there is such that can be embedded into some induced subgraph of , and denote this by Questions:
- (1)
If is a graph with , do we necessarily have ? If not, can we at least acheive for to be ? 2. (2)
Given graphs what is in terms of , where denotes the categorical product? 3. (3)
How (if at all) does relate to the chromatic number ?
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.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Palash Goyal, Emilio Ferrara, Graph Embedding Techniques, Applications, and Performance: A Survey , https://arxiv.org/abs/1705.02801
- 2[2] Hongyun Cai, Vincent W. Zheng, Kevin Chen-Chuan Chang, A Comprehensive Survey of Graph Embedding: Problems, Techniques and Applications , https://arxiv.org/abs/1709.07604
