Let S be an analytically finite Riemann surface of type  that contains at least one puncture x. We can associate to S a complex of curves  that is equipped with a path metric  Let  be the set of mapping classes on S isotopic to the identity on  We show that for any pseudo-Anosov element  and any vertex c in    whenever  and the equality holds for  and for suitable choices of f and c. We also show that for any vertices  that are isotopic to each other on  and satisfy the inequality  there exist infinitely many pseudo-Anosov elements  with their dilatation    and