The 'separation dimension' of a graph G is the smallest natural number k
for which the vertices of G can be embedded in Rk such that any
pair of disjoint edges in G can be separated by a hyperplane normal to one of
the axes. Equivalently, it is the smallest possible cardinality of a family
F of total orders of the vertices of G such that for any two
disjoint edges of G, there exists at least one total order in F
in which all the vertices in one edge precede those in the other. In general,
the maximum separation dimension of a graph on n vertices is Ξ(logn). In this article, we focus on bounded degree graphs and show that the
separation dimension of a graph with maximum degree d is at most
29logβdd. We also demonstrate that the above bound is nearly
tight by showing that, for every d, almost all d-regular graphs have
separation dimension at least βd/2β.Comment: One result proved in this paper is also present in arXiv:1212.675