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arXiv · 2502.14853

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

Wanting Gao·Xudong Chen
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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.

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Wanting Gao, Xudong Chen. 2025-02-20. On the $H$-property for Step-graphons: The Residual Case. https://arxiv.org/abs/2502.14853

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