The H-property of Line Graphons
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 ∞. 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.