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 = (V1 ∪ V2 ∪ V3, E) (|Vi| = n), is hamiltonian, if for any two nonadjacent vertices u ∈ Vi and v ∈ Vj (1 ⩽ i < j ⩽ 3) of G, the following condition is satisfied: |N(u) ∩ Vj| + |N(v) ∩ Vi| ⩾ n + 1.