ar X iv : 1 10 1 . 06 93 v 1 [ m at h . C O ] 4 J an 2 01 1 The C l - free process Lutz Warnke
نویسنده
چکیده
The Cl-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 Cl is created. For every l ≥ 4 we show that, with high probability as n → ∞, the maximum degree is O((n log n)), which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the Cl-free process typically terminates with Θ(n (logn)) 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 nontrivial class of graphsH . 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.
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تاریخ انتشار 2014