2024/09/26 by Fabian Burghart, Burghart, Fabian, Marc Kaufmann +5
Mathematics · #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2409.17764
Consider the random u-uniform hypergraph (or u-graph) process on n vertices, where n is divisible by r>u≥ 2. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete u-graph Kr, it also contains a Kr-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete u-graphs, but for a broader class of so-called nice u-graphs. The purpose of this article is to extend the process coupling for complete u-graphs to the full scope of the sharp threshold result: nice u-graphs. As a byproduct, we obtain the extension of the hitting time result to nice u-graphs. Since the relevant combinatorial bounds in the proof for the Kr-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the Kr-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations.