Separation dimension of sparse graphs

The separation dimension of a graph G is the smallest natural number k for whichthe vertices of G can be embedded in Rk such that any pair of disjoint edges in G canbe separated by a hyperplane normal to one of the axes. Equivalently, it is the smallestpossible cardinality of a family F of permutations of the vertices of G such that for anytwo disjoint edges of G, there exists at least one permutation in F in which all the verticesin one edge precede those in the other. In general, the maximum separation dimension of agraph on n vertices is Θ (log n). In this article, we focus on sparse graphs and show that themaximum separation dimension of a k-degenerate graph on n vertices is O (k log log n) andthat there exists a family of 2-degenerate graphs with separation dimension Ω (log log n).We also show that the separation dimension of the graph G1/2 obtained by subdividingonce every edge of another graph G is at most (1 + o(1)) log logχ(G) where χ(G) is thechromatic number of the original graph.