  In this paper, we study the Potts model on a general graph whose vertices are
partitioned into $m$ blocks of the same size. The interaction between two spins
depends only on whether they belong to the same block or not. We denote the
interaction coefficients for the spins at the different blocks and the same
block by $a, b$, respectively. We show that $J=(b+(m-1)a) / m$ plays the role
of the effective interaction coefficient of the overall system. In
Curie-Weiss-Potts model with $a=0, \ b=1, \ m=1$, critical phase transition
appears at the inverse-temperature $\beta_c$, and mixing transition occurs at
$\beta_s$. Generalizing this fact to our model, we explicitly find the critical
inverse temperatures $\beta_c/J, \ \beta_s/J$ for the phase transition and
mixing, respectively. Enhancing the aggregate path coupling, we prove in a
high-temperature regime $\beta<\beta_s/J$, a cutoff occurs at time
$\large[2(1-2\beta J/q)\large]^{-1}n \log n$ with window size $n$. Moreover,
metastability deduces the exponential mixing in the low-temperature regime
$\beta> \beta_s/J$. These results first show the cutoff of the block Potts
model and suggest a novel extension of the aggregate path coupling.
