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

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 → ∞. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) 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.