= Solution
Use $F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+g f^{abc}A_\mu^bA_\nu^c$ and $\mathcal L=-F_{\mu\nu}^aF^{a\mu\nu}/4$. Its <cubic and quartic Yang-Mills self-interactions> are
$$
\boxed{\mathcal L_3=-\frac g2f^{abc}
(\partial_\mu A_\nu^a-\partial_\nu A_\mu^a)A^{b\mu}A^{c\nu},\qquad
\mathcal L_4=-\frac{g^2}4f^{abc}f^{ade}
A_\mu^bA_\nu^cA^{d\mu}A^{e\nu}.}
$$
Antisymmetry of the <Lie algebra structure constants> lets the cubic expression also be written as $-g f^{abc}(\partial_\mu A_\nu^a)A^{b\mu}A^{c\nu}$.
The cubic vertex has three fields and one <derivative>, corresponding to $n=2,r=1$ in part (b). The quartic vertex has four fields and no <derivatives>, corresponding to $n=4,r=0$. Their dimensions are
$$
[g]=\frac{4-d}{2},\qquad[g^2]=4-d.
$$
With both types present, the same <Feynman diagram> counting gives
$$
D=d-\frac{d-2}{2}E-V_3[g]-V_4[g^2].
$$
In four dimensions both couplings are dimensionless, and the degree is bounded independently of vertex count. Gauge symmetry, implemented consistently after <gauge fixing>, restricts the ultraviolet <counterterms> to the allowed renormalizable gauge-invariant and <gauge fixing>/ghost structures. Thus the counting is consistent with perturbative <renormalizability> of ordinary four-dimensional <Yang-Mills theory>.
In six dimensions $[g]=-1$ and $[g^2]=-2$, giving $D=6-2E+V_3+2V_4$. Increasing the vertex count permits new divergent momentum structures and higher-dimensional operators without a finite closed list of <renormalization> parameters. \b[Ordinary Yang-Mills theory is therefore power-counting renormalizable in four dimensions and nonrenormalizable in six dimensions.] It can still be used in six dimensions as an <effective field theory> with a cutoff and higher-operator expansion; this conclusion concerns the usual perturbative theory.
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