On the H-property for step-graphons and edge polytopes

TL;DR

This paper investigates the conditions under which certain graphons exhibit Hamiltonian decompositions, linking geometric properties of associated edge polytopes to the probabilistic structure of large random graphs.

Contribution

It establishes necessary geometric conditions involving edge polytopes and concentration vectors for the H-property in step-graphons, connecting graph theory with polytope geometry.

Findings

Necessary conditions for the H-property involve full rank of the edge-polytope of the skeleton graph.

The concentration vector must lie within the edge-polytope of the skeleton graph.

Provides a geometric approach to analyzing Hamiltonian decompositions in random graph models.

Abstract

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) $x^* \in \mathcal{X}(S)$.