A sharp threshold for spanning 2‐spheres in random 2‐complexes
Zur Luria, Ran J. Tessler · Proceedings of the London Mathematical Society · 2019
A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to a d-dimensional sphere. We consider the Linial–Meshulam model for random simplicial complexes, and prove that there is a sharp threshold at p = e / γ n for the appearance of a Hamiltonian 2-sphere in a random 2-complex, where γ = 4 4 / 3 3 .