Given a graph $(V,E)$, I'm interested in embedding it into a Euclidean space $\mathbb{R}^n$ such that each vertex $v\in V$ becomes a point $x_v\in\mathbb{R}^n$ and $d(x_v,x_u) \leq 1$ (Euclidean distance) iff $(v,u) \in E$.

Is this embedding always possible given high enough dimension $n$? If so, given a graph, can we estimate the upper bound for $n$ (as a function of $|V|$, or something similar) such that the above embedding is possible?


1 Answer 1


Your parameter is known as sphericity, first defined by Maehara, Space graphs and sphericity.

Maehara showed that every graph has such an embedding. Given a graph $G = (V,E)$, embed $x \in V$ into the $V$-indexed vector $v_x$ given by $v_x(x) = M$, $v_x(y) = 1$ if $(x,y) \in E$, and $v_x(y) = 0$ if $(x,y) \notin E$.

If $(x,y) \notin E$ then $\|v_x - v_y\|^2 \geq 2M^2$, considering coordinates $x,y$.

In contrast, if $(x,y) \in E$ then $\|v_x - v_y\|^2 \leq 2(M-1)^2 + |V|-2$.

When $M$ is large enough, $2M^2 > 2(M-1)^2 + |V|-2$, and so we can normalize the vectors so that they satisfy your condition.

The minimum dimension needed to embed a graph is known as its sphericity.

Maehara, Dispersed points and geometric embedding of complete bipartite graphs later showed that the sphericity of the complete bipartite graph $K_{n,n}$ is between $n$ and $\tfrac32 n$.

Some other results are cited in Bilu and Linial, Monotone maps, sphericity and bounded second eigenvalue.


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