Separation in Graphs: A Survey and Some Recent Results

Let G and H be graphs, with |G| ≤ |H|. Consider a one-to-one map f:G→H, and define |f| = min{distH(f(x),f(y)): xy E(G)}. Now let the separation of G in H be defined by sep(G,H) = max{|f|: all one-to-one maps f:G→H}. We survey some results of the past several years on this parameter, in particular using Alon and Milman's results on the connection between the second largest eigenvalue of a graph and isoperimetric inequalities to obtain bounds for sep(G,Q(n)), where Q(n) is the n-dimensional hypercube and G is arbitrary. Also included are two new results, one involving the Kruskal-Katona theorem as it applies to separating complete bipartite graphs into hypercubes, the other being an algorithmic solution to separating complete graphs into trees.