  We consider the problem of sampling from the ferromagnetic Potts and
random-cluster models on a general family of random graphs via the Glauber
dynamics for the random-cluster model. The random-cluster model is parametrized
by an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establish
that for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal
$\Theta(n\log n)$ steps on $n$-vertex random graphs having a prescribed degree
sequence with bounded average branching $\gamma$ throughout the full
high-temperature uniqueness regime $p<p_u(q,\gamma)$.
  The family of random graph models we consider includes the Erd\H{o}s--R\'enyi
random graph $G(n,\gamma/n)$, and so we provide the first polynomial-time
sampling algorithm for the ferromagnetic Potts model on Erd\H{o}s--R\'enyi
random graphs for the full tree uniqueness regime. We accompany our results
with mixing time lower bounds (exponential in the largest degree) for the Potts
Glauber dynamics, in the same settings where our $\Theta(n \log n)$ bounds for
the random-cluster Glauber dynamics apply. This reveals a novel and significant
computational advantage of random-cluster based algorithms for sampling from
the Potts model at high temperatures.
