When does the K₄‐free process stop?

Abstract

The K4‐free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K4. Let G be the random maximal K4‐free graph obtained at the end of the process. We show that for some positive constant C, with high probability as n→∞, the maximum degree in G is at most Cn3/5logn5. This resolves a conjecture of Bohman and Keevash for the K4‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has Θ(n8/5logn5) edges and is ‘nearly regular’, i.e., every vertex has degree Θ(n3/5logn5). This answers a question of Erdős, Suen and Winkler for the K4‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least Ω(n2/5(logn)4/5/loglogn), which matches an upper bound obtained by Bohman up to a factor of Θ(loglogn). Our analysis of the K4‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erdős and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014

A note on the random greedy independent set algorithm

“…Bohman and Keevash conjecture that this is the case for the H ‐free process when H is strictly 2‐balanced, but even this remains widely open. The conjecture has been verified in some special cases, including the K 3 ‐free process , the K 4 ‐free process and the ‐free process for all .…”

“…It is tempting to speculate that the lower bound in Theorem 1.1 gives the correct order of magnitude of the maximal independent set produced by the random greedy independent set algorithm for a broad class of hypergraphs H. Bohman and Keevash conjecture that this is the case for the H-free process when H is strictly 2-balanced, but even this remains widely open. The conjecture has been verified in some special cases, including the K 3 -free process [3], the K 4 -free process [26,27] and the C -free process for all ≥ 4 [21,22,25].…”

The diamond‐free process

“…First, we have replaced the condition (1). The necessity of this condition was first pointed out by Warnke (see the appendix of [15]). Next, we have removed three error functions, e , γ and h from our statement by setting e (t) = 0 and γ (t) = θ (t) − θ (0), h (t) = θ (t), and requiring θ (t) is smooth and nonnegative.…”

“…This decision was inspired by [15], to which we refer the reader for a generalization of Bohman and Keevash's lemma in which the number of variable types can grow with n.…”

“…Comparing with Lemma 7.3 of [2], we have made a few minor notation changes, including the indexing of the variable types by a fixed finite set L rather than a set of integers. This decision was inspired by [15], to which we refer the reader for a generalization of Bohman and Keevash's lemma in which the number of variable types can grow with n.…”