  The question of a modification of the running gauge coupling of (non-)
abelian gauge theories by an incorporation of the quantum gravity contribution
has recently attracted considerable interest. In this letter we perform an
involved diagrammatical calculation in the full Einstein-Yang-Mills system both
in cut-off and dimensional regularization at one loop order. It is found that
all gravitational quadratic divergencies cancel in cut-off regularization and
are trivially absent in dimensional regularization so that there is no
alteration to asymptotic freedom at high energies. The logarithmic divergencies
give rise to an extended effective Einstein-Yang-Mills Lagrangian with a
counterterm of dimension six. In the pure Yang-Mills sector this counterterm
can be removed by a nonlinear field redefinition of the gauge potential,
reproducing a classical result of Deser, Tsao and van Nieuwenhuizen obtained in
the background field method with dimensional regularization.
