  Graphons $W$ can be used as stochastic models to sample graphs $G_n$ on $n$
nodes for $n$ arbitrarily large. A graphon $W$ is said to have the $H$-property
if $G_n$ admits a decomposition into disjoint cycles with probability one as
$n$ goes to infinity. Such a decomposition is known as a Hamiltonian
decomposition. In this paper, we provide necessary conditions for the
$H$-property to hold. The proof builds upon a hereby established connection
between the so-called edge polytope of a finite undirected graph associated
with $W$ and the $H$-property. Building on its properties, we provide a purely
geometric solution to a random graph problem. More precisely, we assign two
natural objects to $W$, which we term concentration vector and skeleton graph,
denoted by $x^*$ and $S$ respectively. We then establish two necessary
conditions for the $H$-property to hold: (1) the edge-polytope of $S$, denoted
by $\mathcal{X}(S)$, is of full rank, and (2) $x^* \in \mathcal{X}(S)$.
