A problem on claw-free homogeneously traceable

In 1988 Faudree et al. proved that let G be a 3-connected K_1,3-free graph of order n, if |N(x)∪ N(y)|≥(2n-4)/3 for each pair of nonadjacent vertices x,y, then G is Homogeneously traceable. In 1991nian Bauer et al. proved that let G be a 3-connected K_1,3-free graph of order n, if |N(x) ∪ N(y)|≥(2n-5)/3 for each pair of nonadjacent vertices x,y, then G is traceable. In this note we prove the further result: let G be a 3-connected K_1,3-free graph of order n, if |N(x) ∪ N(y)|≥(2n-6)/3 for each pair of nonadjacent vertices x,y with 1≤|N(x) ∩ N(y)|≤α-1, then G is Homogeneously traceable.