  We propose a new reformulation of Yang-Mills theory in which three- and
four-gluon self-interactions are eliminated at the price of introducing a
sufficient number of auxiliary fields. We discuss the validity of this
reformulation in the possible applications such as dynamical gluon mass
generation, color confinement and glueball mass calculation. Moreover, we set
up a new $1/N_c$ color expansion in the $SU(N_c)$ Yang-Mills theory based on
this reformulation. In fact, we give the Feynman rules of the $1/N_c$ expansion
in the manifestly Lorentz covariant gauge. The Yang-Mills theory is defined on
a non-trivial vacuum where color--singlet transverse gluon pair condensations
take place by the attractive gluonic self-interactions. This vacuum
condensation provides a common non-vanishing mass for all the gluons with color
symmetry being preserved. It is shown that the auxiliary fields become
dynamical by acquiring the kinetic term due to quantum corrections. Then the
static potential between a pair of color charges is derived as a combination of
the Yukawa-type potential and the linear potential with non-vanishing string
tension. The mass of the lightest scalar glueball is calculated as the ratio to
the gluon mass. The explicit calculations are performed as a partial
resummation of the leading order diagrams for the small 't Hooft coupling.
