The C ℓ ‐free process
2012; Wiley; Volume: 44; Issue: 4 Linguagem: Inglês
10.1002/rsa.20468
ISSN1098-2418
Autores Tópico(s)Markov Chains and Monte Carlo Methods
ResumoAbstract The ‐free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of is created. For every we show that, with high probability as , the maximum degree is , which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the ‐free process typically terminates with edges, which answers a question of Erdős, Suen and Winkler. This is the first result that determines the final number of edges of the more general H ‐free process for a non‐trivial class of graphs H . We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to ‘transfer’ certain decreasing properties from the binomial random graph to the H ‐free process. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 490–526, 2014
Referência(s)