Abstract: In 1998 a conjecture was suggested for the conference of Graph theory, combinatorics, and applications at Kalamazooin USA as follows： let G be a 3-connected K1.3-free graph of order n, if ｜N（x） ∪N（y） ｜≥（2n-6）/3 for each pair of nonadjacent vertices x,y, t

Key words: K1,3-free graphs, Neighborhood unions, Generalizing neighborhood unions,Hamiltonian