  We explore in this paper sufficient conditions for the $H$-property to hold,
with a particular focus on the so-called line graphons. A graphon is a
symmetric, measurable function from the unit square $[0,1]^2$ to the closed
interval $[0,1]$. Graphons can be used to sample random graphs, and a graphon
is said to have the $H$-property if graphs on $n$ nodes sampled from it admit a
node-cover by disjoint cycles -- such a cover is called a Hamiltonian
decomposition -- almost surely as $n \to \infty$. A step-graphon is a graphon
which is piecewise constant over rectangles in the domain. To a step-graphon,
we assign two objects: its concentration vector, encoding the areas of the
rectangles, and its skeleton-graph, describing their supports. These two
objects were used in our earlier work [3] to establish necessary conditions for
a step-graphon to have the $H$-property. In this paper, we prove that these
conditions are essentially also sufficient for the class of line-graphons,
i.e., the step-graphons whose skeleton graphs are line graphs with a self-loop
at an ending node. We also investigate borderline cases where neither the
necessary nor the sufficient conditions are met.
