The H-property of Line Graphons

TL;DR

This paper investigates the conditions under which line graphons possess the H-property, demonstrating that certain necessary conditions are also sufficient for this class, and explores borderline cases.

Contribution

It proves that the necessary conditions for the H-property are also sufficient for line graphons, extending previous work on step-graphons.

Findings

Necessary conditions are also sufficient for line graphons.

Characterization of the H-property for line graphons.

Analysis of borderline cases where conditions are not met.

Abstract

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.