Title: Sharp thresholds for spanning regular subgraphs
Abstract: A graph property is called increasing if it is closed under addition of edges. For an increasing property, the probability of its occurrence in the binomial random graph G(n,p) transitions rapidly (in asymptotics) from 0 to 1 as p crosses the so-called probability threshold. Since the original paper of Erdős and Rényi the task of determining the asymptotic behaviour of threshold probabilities for increasing properties has been a central topic in probabilistic combinatorics. While the asymptotic order of the probability threshold has been determined for many natural increasing graph properties, a general solution remains unknown, and determining the exact asymptotics is even more challenging. In the talk I will provide an answer to the latter question for a class of increasing properties generated by d-regular graphs. This family of d-regular graphs contains asymptotically almost all d-regular graphs and, in particular, it contains the square of a cycle. This resolves a conjecture of Kahn, Narayanan, and Park, regarding the asymptotics of the threshold for the appearance of the square of a Hamilton cycle.