  The "separation dimension" of a graph $G$ is the minimum positive integer $d$
for which there is an embedding of $G$ into $\mathbb{R}^d$, such that every
pair of disjoint edges are separated by some axis-parallel hyperplane. We prove
a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every
graph with maximum degree $\Delta$ has separation dimension less than
$20\Delta$, which is best possible up to a constant factor. We also prove that
graphs with separation dimension 3 have bounded average degree and bounded
chromatic number, partially resolving an open problem by Alon et al. [J. Graph
Theory 2018].
