THE COORDINATE REPRESENTATION OF A GRAPH AND N-UNIVERSAL GRAPH OF RADIUS-1

Every graph can be represented as the intersection graph on a family o f closed unit cubes in Euclidean space E(n). Cube vertices have intege r coordinates. The coordinate matrix, A(G)={v(nk)} of a graph G is def ined by the set of cube coordinates. The imbedded dimension of a graph , Bp(G), is a number of columns in matrix A(G) such that each of them has at least two distinct elements v(nk) not-equal v(pk). We show that Bp(G)=cub(G) for some graphs, and Bp(G) less-than-or-equal-to n-2 for any graph G on n vertices. The coordinate matrix uses to obtain the g raph U of radius 1 with 3n-2 vertices that contains as an induced subg raph a copy of any graph on n vertices.