A neighborhood condition which implies the existence of a complete multipartite subgraph

The complete multipartite graph with d partite sets, each containing m vertices, will be denoted by K(d; m). If all of the partite sets are not of the same cardinality, the graph K(mi, m2, . . . . md) is the complete d-partite graph with m, vertices in each partite sets, 1 <i<d. The special case where mi= 1 for every i, yields the complete graph on d vertices and is denoted by Kd. The join of graphs G1 and G2, denoted by G1+G2, has V(G,+G2)= V(G,)uV(G,) and E(G1+G2)=E(G1)uE(G2)u{~u~~ is an element of V(G,), v is an element of V(G,)}. Thus, for example, if G=K,+K(3; m),