The C_ℓ‐free process

Abstract

Abstract. The C ℓ -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 ℓ is created. For every ℓ ≥ 4 we show that, with high probability as n → ∞, the maximum degree is O((n log n) 1/(ℓ−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 ℓ -free process typically terminates with Θ(n ℓ/(ℓ−1) (log n) 1/(ℓ−1) ) 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 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. IntroductionThe random graph process was introduced by Erdős and Rényi [10] in 1959. It starts with the empty graph on n vertices and adds new edges one by one, where each edge is chosen uniformly at random among all edges not yet present. Since then it has been studied extensively, and many tools and methods for investigating its typical properties have been developed, see e.g. [5,8,12]. In this work we consider a natural variant of the above process which has very recently received a considerable amount of attention [2,3,11,17,18,23,24,25,26,27].The H-free process was suggested by Bollobás and Erdős [4] in 1990, as a way to generate an interesting probability distribution on the set of maximal H-free graphs with potential applications to Ramsey Theory. Given some fixed graph H, it is a modification of the classical random graph process, where each new edge is chosen uniformly at random subject to the condition that no copy of H is formed. It was first described in print in 1995 by Erdős, Suen and Winkler [9], who asked how many edges the final graph typically has (this also appears as a problem in [7]). The main difficulty when analysing this process is that there is a complicated dependence among the edges; the order in which they are inserted is also relevant.The first results addressed certain special graphs, determining the typical final number of edges up to logarithmic factors. The case H = C 3 was studied in 1995 by Erdős, Suen and Winkler [9], and in 2000 Bollobás and Riordan [6] considered H ∈ {K 4 , C 4 }. In fact, a result of Ruciński and Wormald [21] predates those mentioned above: in 1992 they considered the (much simpler) maximum degree d-process, which corresponds to the case H = K 1,d+1 , and showed that whp 1 it ends with ⌊nd/2⌋ edges. The general H-free process was first analysed independently by Bollobás and Riordan [6] and Osthus and Taraz [16] in 2000. In fact, they assumed that H satisfies a certain density condition (strictly 2-...

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].…”

On the Method of Typical Bounded Differences

“…Matching bounds up to constant factors have only been established for some special forbidden graphs and the class of C ℓ -free processes, see e.g. [44,45]. The final graph of the K s -free process also yields the best known lower bounds on the Ramsey numbers R(s, t) with s ≥ 4, see [4,6].…”

“…In this case Lemma 26 remains true if we modify D to at most, say, Ψ H = (log n)n vH −2 p eH −1 copies, and so the coupling of Lemma 27 carries over (it only uses I). With (44) in mind, Theorem 29 shows that the expected final number of edges is µ = Θ(n 2−1/m2(H) ). Adjusting the proof of Theorem 28 with c k = 2e H Ψ H , a short calculation shows that we obtain concentration on an interval of length µn −γ with γ = γ(H) > 0 whenever v H ≥ 4 and m 2 (H) < (2e H − 3)/(2v H − 6) or v H = 3 and e H ≥ 2.…”

Packing Nearly Optimal Ramsey R(3,t) Graphs

“…We believe that variants of Theorems 1 and 4 also hold for many other forbidden graphs (using semirandom variants of the H-free process [25,6,34,35,27]); we hope to return to this topic in a future work.…”