  A graphon satisfies the $H$-property if graphs sampled from it contain a
Hamiltonian decomposition almost surely, which in turn implies that the
corresponding network topologies are, e.g., structurally stable and
structurally ensemble controllable. In recent papers, we have exhibited a set
of conditions that is essentially necessary and sufficient for the $H$-property
to hold for the finite-dimensional class of step-graphons. The extension to the
infinite-dimensional case of general graphons was hindered by the fact that
said conditions relied on objects that do not admit immediate extensions to the
infinite-dimensional case. We outline here our approach to bypass this
difficulty and state conditions that guarantee that the $H$-property holds for
general graphons.
