The C_\ell-free process

TL;DR

This paper analyzes the C_free process, establishing bounds on maximum degree and final edge count, confirming conjectures, and introducing a novel method for transferring properties from random graphs to H-free processes.

Contribution

It provides the first determination of the final number of edges for a non-trivial H-free process and introduces a new transfer technique for properties from random graphs.

Findings

Maximum degree is O((n log n)^{1/(-1)}) with high probability.

Final number of edges is (n^{/(-1)}( log n)^{1/(-1)}) asymptotically.

Verified a conjecture on average degree and improved bounds on independence number.

Abstract

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 \emph{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.