  The quantum Bruhat graph, which is an extension of the graph formed by
covering relations in the Bruhat order, is naturally related to the quantum
cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing
that the minimal monomial in the quantum parameters that occurs in the quantum
product of two Schubert classes has a simple interpretation in terms of
directed paths in this graph.
  We define path Schubert polynomials, which are quantum cohomology analogues
of skew Schubert polynomials recently introduced by Lenart and Sottile. They
are given by sums over paths in the quantum Bruhat graph of type A. The 3-point
Gromov-Witten invariants for the flag manifold are expressed in terms of these
polynomials. This construction gives a combinatorial description for the set of
all monomials in the quantum parameters that occur in the quantum product of
two Schubert classes.
