Hamiltonicity of Step-graphon

Abstract

A step-graphon has the strong (resp., weak) $H$-property if a directed, random graph sampled from it has a Hamilton cycle (resp., a node-wise disjoint cycle cover) asymptotically almost surely. The weak/strong $H$-property is essentially a zero-one property. We identify key objects associated with the step-graphon that matter for the zero-one law and provide a complete characterization.