Random triangle removal

By Tom Bohman, Alan M. Frieze, Eyal Lubetzky. 2012

61 citations recorded by the source corpus.

Starting from a complete graph on $n$ vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollob\'as and Erd\H{o}s (1990) conjectured that the expected final number of edges has order $n^{3/2}$, motivated by the study of the Ramsey number $R(3,t)$. An upper bound of $o(n^2)$ was shown by Spencer (1995) and independently by R\"odl and Thoma (1996). Several bounds were given for variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald (1999)), while the best known upper bound for the original question of Bollob\'as and Erd\H{o}s was $n^{7/4+o(1)}$ due to Grable (1997). No nontrivial lower bound was available. Here we prove that with high probability the final number of edges in random triangle removal is equal to $n^{3/2+o(1)}$, thus confirming the 3/2 exponent conjectured by Bollob\'as and Erd\H{o}s and matching the predictions of Spencer et al. For the upper bound, for any fixed $\epsilon>0$ we construct a family of $\exp(O(1/\epsilon))$ graphs by gluing $O(1/\epsilon)$ triangles sequentially in a prescribed manner, and dynamically track all homomorphisms from them, rooted at any two vertices, up to the point where $n^{3/2+\epsilon}$ edges remain. A system of martingales establishes concentration for these random variables around their analogous means in a random graph with corresponding edge density, and a key role is played by the self-correcting nature of the process. The lower bound builds on the estimates at that very point to show that the process will typically terminate with at least $n^{3/2-o(1)}$ edges left.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bfe56f8e191f3ed28928d7cb04ac627709383262

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Counting designs (2015, methodology): “It is shown in [2] that whp the process persists until only O(n3/2+o(1)) edges remain, but we will stop at n2−10 −7 edges (i.”
The triangle-free process and R(3,k) (2013, methodology): “showing that (with high probability) they closely follow the solution of a corresponding family of dierential equations. This method has been used with great success in recent years, see for example [8, 10, 12, 13]; we note in particular the recent result of Bohman, Frieze and Lubetzky [12] that the”
Almost all Steiner triple systems have perfect matchings (2016, methodology): “The triangle removal process starting from the complete graph has already been thoroughly analysed; see in particular the precise analysis by Bohman, Frieze and Lubetzky [6] and their simplified analysis in [5].”