"The C_ell -free process".

"The C_ell -free process".

Tue, 08/02/2011
16:30 		Lutz Warnke Combinatorial Theory Seminar Add to calendar SR2
"The C_ell -free process".
The $ C_\ell $-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 $ C_\ell $ is created. For every $ \ell \geq 4 $ we show that, with high probability as $ n \to \infty $, the maximum degree is $ O((n \log n)^{1/(\ell-1)}) $, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the $ C_\ell $-free process typically terminates with $ \Theta(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)}) $ edges, which answers a question of Erd\H{o}s, 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.