On the euclidean dimension of a complete multipartite graph

The euclidean dimension of a graph G, e(G) , is the minimum n such that the vertices of G can be placed in euclidean n -space, R n , in such a way that adjacent vertices have distance 1 and nonadjacent vertices have distances other than 1. Let G = K ( n 1 ,…, n s+1+u ) be a complete (s+1+u)-partite graph with vertex-classes consisting of s sets of size 1, t sets of size 2, and u sets of size ⩾3. We prove that e(G) = s + t + 2 u if t + u ⩾2, and e(G) = s + t + 2 u - 1 if t + u ⩽1.