On the $H$-property for Step-graphons: The Residual Case

TL;DR

This paper studies the probability of Hamiltonian decompositions in graphs sampled from step-graphons, especially in the residual case where the zero-one law does not hold, providing explicit formulas and numerical validation.

Contribution

It extends the understanding of the $H$-property in step-graphons by analyzing the residual case and deriving explicit probability limits.

Findings

Probability limit exists in the residual case.

Explicit expression for the probability limit.

Numerical validation supports theoretical results.

Abstract

We investigate the $H$-property for step-graphons. Specifically, we sample graphs $G_n$ on $n$ nodes from a step-graphon and evaluate the probability that $G_n$ has a Hamiltonian decomposition in the asymptotic regime as $n\to\infty$. It has been shown that for almost all step-graphons, this probability converges to either zero or one. We focus in this paper on the residual case where the zero-one law does not apply. We show that the limit of the probability still exists and provide an explicit expression of it. We present a complete proof of the result and validate it through numerical studies.