\in \Omega  .\end{aligned}$$ Under *path coupling*, the coupling is only defined on a subset $\Lambda $ of $ \Omega \times \Omega$. This restricted coupling is then extended to a coupling on the whole of $\Omega \times \Omega$ along paths in the state space $\Omega$. In our setting, we have $\Omega = [q]^V$, where $V$ is the vertex set of some fixed graph. For $\sigma, \sigma' \in \Omega$, we write $d(\sigma, \sigma')$ for the number of vertices on which $\sigma$ and $\sigma'$ differ in colour (that is, the Hamming distance). Define $\Lambda \subseteq \Omega \times \Omega$ by $$\Lambda = \{(\sigma,\sigma'): d(\sigma,\sigma')=1 \}.$$ The key property of $\Lambda$ required for the path coupling method is that for any $\sigma,\sigma'\in\Omega$, by recolouring the $d(\sigma,\sigma')$ disagreeing vertices one by one in an arbitrary order, we obtain a path of length $d(\sigma,\sigma')$ from $\sigma$ to $\sigma'$, with consecutive elements of the path corresponding to an element of $\Lambda$.

\[See [@DG98] for example\] \[le:coup\] Let $\Omega=[q]^V$ and $\Lambda$ be as above, with $n:=|V|$, and let $\mathcal{M}$ be some Markov chain on $\Omega$. Suppose that we can define a coupling $(A,B)\mapsto (A',B')$ for $\mathcal{M}$ on $\Lambda$ such that for some constant $\beta < 1$ and all $(A,B)\in 