  We investigate the existence or non-existence of spectral minimal partitions
of unbounded metric graphs, where the operator applied to each of the partition
elements is a Schr\"odinger operator of the form $-\Delta + V$ with suitable
(electric) potential $V$, which is taken as a fixed, underlying ``landscape''
on the whole graph.
  We show that there is a strong link between spectral minimal partitions and
infimal partition energies on the one hand, and the infimum $\Sigma$ of the
essential spectrum of the corresponding Schr\"odinger operator on the whole
graph on the other. Namely, we show that for any $k\in\mathbb{N}$, the infimal
energy among all admissible $k$-partitions is bounded from above by $\Sigma$,
and if it is strictly below $\Sigma$, then a spectral minimal $k$-partition
exists. We illustrate our results with several examples of existence and
non-existence of minimal partitions of unbounded and infinite graphs, with and
without potentials.
  The nature of the proofs, a key ingredient of which is a version of Persson's
theorem for quantum graphs, strongly suggests that corresponding results should
hold for Schr\"odinger operator-based partitions of unbounded domains in
Euclidean space.
