  In this paper we analyze the one-loop renormalization of the
$\theta$-expanded $\rm SU(N)$ Yang-Mills theory. We show that the {\it freedom
parameter} $a$, key to renormalization, originates from higher order
non-commutative gauge interaction, represented by a higher derivative term $ b
h \theta^{\mu\nu}\hat F_{\mu\nu}\star\hat F_{\rho\sigma}\star\hat
F^{\rho\sigma}$. The renormalization condition fixes the allowed values of the
parameter $a$ to one of the two solutions: $a=1$ or $a=3$, i.e. to $b=0$ or to
$b=1/2$, respectively. When the higher order interaction is switched on,
($a=3$), pure non-commutative SU(N) gauge theory at first order in
$\theta$-expansion becomes one-loop renormalizable for various representations
of the gauge group. We also show that, in the case $a=3$ and the adjoint
representation of the gauge fields, the non-commutative deformation parameter
$h$ has to be renormalized and it is asymptotically free.
