On the H-Property for Step-Graphons and Edge Polytopes

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 letter, 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 maximal rank, and (2)
$x^{*}$
belongs to
$\mathcal {X}{(}S{)}$
.