  We introduce edge labeled Young tableaux. Our main results provide a
corresponding analogue of [Sch\"{u}tzenberger '77]'s theory of jeu de taquin.
These are applied to the equivariant Schubert calculus of Grassmannians.
Reinterpreting, we present new (semi)standard tableaux to study factorial Schur
polynomials, after [Biedenharn-Louck '89], [Macdonald '92] and [Goulden-Greene
'94] and others.
  Consequently, we obtain new combinatorial rules for the Schubert structure
coefficients, complementing work of [Molev-Sagan '99], [Knutson-Tao '03],
[Molev '08] and [Kreiman '09]. We also describe a conjectural generalization of
one of our rules to the equivariant K-theory of Grassmannians, extending work
of [Thomas-Yong '07]. This conjecture concretely realizes the "positivity"
known to exist by [Anderson-Griffeth-Miller '08]. It provides an alternative to
the conjectural rule of Knutson-Vakil reported in [Coskun-Vakil '06].
