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The paper concerns the $q$-state Potts model (i.e., with spin values in$\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e.,with $k+1$ edges emanating from each vertex) in an external (possibly random)field. We construct the so-called splitting Gibbs measures (SGM) usinggeneralized boundary conditions on a sequence of expanding balls, subject to asuitable compatibility criterion. Hence, the problem of existence/uniqueness ofSGM is reduced to solvability of the corresponding functional equation on thetree. In particular, we introduce the notion of translation-invariant SGMs andprove a novel criterion of translation invariance. Assuming a ferromagneticnearest-neighbour spin-spin interaction, we obtain various sufficientconditions for uniqueness. For a model with constant external field, we providein-depth analysis of uniqueness vs.\ non-uniqueness in the subclass ofcompletely homogeneous SGMs by identifying the phase diagrams on the"temperature--field" plane for different values of the parameters $q$ and $k$.In a few particular cases (e.g., $q=2$ or $k=2$), the maximal number ofcompletely homogeneous SGMs in this model is shown to be $2^q-1$, and we make aconjecture (supported by computer calculations) that this bound is valid forall $q\ge 2$ and $k\ge2$.

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field