A contact pattern is defined as a graph with the vertex set corresponding to the set of nonoverlapping spheres in Euclidean space, where two vertices are adjacent when corresponding spheres touch each other. One can see that any finite graph G is a contact pattern in some n-space \(E^ n\). The question is to determine the contact dimension of a graph which is defined as the smallest n such that G is a contact pattern in \(E^ n\). In the article are presented the exact values of the contact dimension of complete multipartite graphs and cubes. There are proved some estimations for trees and other special classes of graphs.