= Solution
{c}
The <proper quark-gluon vertex colour factors> come from two <one-particle-irreducible Feynman diagrams>, shown in the upper-middle and upper-right panels. One has a virtual <gluon> joining the open <quark> line on opposite sides of the external <gluon> insertion. The other has two virtual <gluons> meeting the external <gluon> at a three-gluon <Feynman vertex>, with their other ends joined by an internal <quark>. A <Faddeev-Popov ghost field> has no direct <quark> coupling, so it supplies no additional one-loop <Feynman vertex> graph here.
Let $T^a$ be the Hermitian generators of the <fundamental representation> of $SU(N)$, with $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, $[T^a,T^b]=if^{abc}T^c$, $\sum_bT^bT^b=C_F\mathbf1$, and $f^{abc}f^{dbc}=C_A\delta^{ad}$. Here $C_F=(N^2-1)/(2N)$ and $C_A=N$ are <quadratic Casimir operators> evaluated in the <fundamental representation> and the <Adjoint representation>.
For the first graph, the ordered product is $\sum_bT^bT^aT^b$. Expand the double <commutator>:
$$
\sum_b[T^b,[T^b,T^a]]=2C_FT^a-2\sum_bT^bT^aT^b=C_AT^a.
$$
Thus its <colour factor> is
$$
\boxed{\sum_bT^bT^aT^b=\left(C_F-\frac{C_A}{2}\right)T^a=-\frac1{2N}T^a=-\frac16T^a\quad(N=3).}
$$
For the second graph choose the orientation of the three-gluon <Feynman vertex> so that its group tensor is $f^{abc}$ and the ordered product along the <quark> line is $T^bT^c$. Antisymmetry removes the <anticommutator> part:
$$
\boxed{f^{abc}T^bT^c=\frac12f^{abc}[T^b,T^c]=\frac{iC_A}{2}T^a=\frac{3i}{2}T^a\quad(N=3).}
$$
The corresponding real group coefficient is $C_A/2=3/2$ after the conventional factor of $i$ is incorporated into the remaining <Feynman rules>. If the vertex is written with $if^{abc}$, its raw matrix product is instead $-C_AT^a/2$. Reversing the ordered product reverses its sign together with the orientation-dependent three-gluon <Feynman rule>; these conventions describe the same <scattering amplitude>. External-leg corrections and local <counterterms> are separate from these two proper one-loop graphs.
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