= Solution
Write $G=G^{(0)}(\tau-\tau')$ and $g_0=G^{(0)}(0)$. With two identical degree-$n$ <vertices>, let $r$ lines join them. The remaining $n-r$ half-edges on each <vertex> form self-loops. Both $n$ and $r$ are even. The number of <Wick contractions> is
$$
N_{n,r}=\binom nr^2r!\bigl[(n-r-1)!!\bigr]^2,
$$
since one chooses and matches the joining half-edges and then pairs the rest independently. Dividing by the expansion denominator $2!(n!)^2$ gives every <symmetry factor> weight without guessing graph multiplicities.
For two quartic <vertices>, $r=0,2,4$ gives
$$
\boxed{\left.\frac Z{Z_0}\right|_{\lambda^2}
=\lambda^2\int_0^\beta d\tau\,d\tau'\left[\frac{g_0^4}{128}+\frac{g_0^2G^2}{16}+\frac{G^4}{48}\right].}
$$
The graphs are respectively the product of two first-order two-loop graphs; two <vertices> joined by two lines with one self-loop on each; and two <vertices> joined by four lines with no self-loops.
For two sextic <vertices>, $r=0,2,4,6$ gives
$$
\boxed{\left.\frac Z{Z_0}\right|_{\mu^2}
=\mu^2\int_0^\beta d\tau\,d\tau'\left[\frac{g_0^6}{4608}+\frac{g_0^4G^2}{256}
+\frac{g_0^2G^4}{192}+\frac{G^6}{1440}\right].}
$$
The corresponding numbers of self-loops on each <vertex> are $3,2,1,0$. All seven second-order graphs are drawn in <3/e/solution>; in each series the $r=0$ graph is disconnected and the others are connected.
For completeness, a mixed quartic-sextic pair, not needed for the two pure second-order coefficients, contributes
$$
\left.\frac Z{Z_0}\right|_{\lambda\mu}=\lambda\mu\int_0^\beta d\tau\,d\tau'
\left[\frac{g_0^5}{384}+\frac{g_0^3G^2}{32}+\frac{g_0G^4}{48}\right].
$$
Taking $\log Z$ removes the products of first-order graphs, that is, the $r=0$ terms. The requested <partition function> retains them.
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