A degree sum condition on Hamiltonian cycles in balanced 3-partite graphs

A k-partite graph in which each partite set has the same number of vertices is said to be a balanced k-partite graph. We show that a balanced 3-partite graph G = (V-1 boolean OR V-2 boolean OR V-3, E) (\V-i\ = n), is hamiltonian, if for any two nonadjacent vertices u is an element of V-i and v is an element of V-j (1 less than or equal to i < j less than or equal to 3) of G, the following condition is satisfied: \N(u) boolean AND V-j\ + \N(v) boolean AND V-i\ greater than or equal to n + 1. (C) 1998 Elsevier Science B.V.