  We address the question of the existence of quantum channels that are
divisible in two quantum channels but not in three or, more generally, channels
divisible in $n$ but not in $n+1$ parts. We show that for the qubit those
channels \textit{do not} exist, whereas for general finite-dimensional quantum
channels the same holds at least for full Kraus rank channels. To prove these
results, we introduce a novel decomposition of quantum channels which separates
them into a boundary and Markovian part, and it holds for any finite dimension.
Additionally, the introduced decomposition amounts to the well-known connection
between divisibility classes and implementation types of quantum dynamical
maps, and can be used to implement quantum channels using smaller quantum
registers.
