Separation Dimension And Degree

The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into Double-struck capital 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].