  We consider compact homogeneous spaces G/H of positive Euler characteristic
endowed with an invariant almost complex structure J and the canonical action
\theta of the maximal torus T ^{k} on G/H. We obtain explicit formula for the
cobordism class of such manifold through the weights of the action \theta at
the identity fixed point eH by an action of the quotient group W_G/W_H of the
Weyl groups for G and H. In this way we show that the cobordism class for such
manifolds can be computed explicitly without information on their cohomology.
We also show that formula for cobordism class provides an explicit way for
computing the classical Chern numbers for (G/H, J). As a consequence we obtain
that the Chern numbers for (G/H, J) can be computed without information on
cohomology for G/H. As an application we provide an explicit formula for
cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n,
Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular
interesting examples.
