Abstract: In this paper, let u, v, w be arbitrary vertices in simple graph of order n-G=(V,E) satisfing wu ∈V(G),d(u,v)=d(v,w)=2, with|N(u) ∪N(v)|+d(w) ≥ n, then G is Hamilto-nian. The result is better than those of other authors.

Key words: Hamilton path, Hamilton graph, sufficient condition, cycles