Under the assumption that G is let to be a 3-connected graph of ordern(n≥11),it has proved that,if for all vertices at an interval of 2 in between(i.e. ,d(u)+d(v)≥n-1 or|N(u∩N(v)|α+1 or|(u)∪N(v)|≥n-δ+1 holds,then G is Hamilton-connected unless G belongs to some special groups of graphs.