Random triangle removal

Tom Bohman, Alan Frieze, Eyal Lubetzky

Research output: Contribution to journalArticle

Abstract

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). Bollobás and Erdos (1990) conjectured that the expected final number of edges has order n<sup>3/2</sup>. An upper bound of o(n<sup>2</sup>) was shown by Spencer (1995) and independently by Rödl 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 Bollobás and Erdos was n<sup>7/4+o(1)</sup> 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<sup>3/2+o(1)</sup>, thus confirming the 3/2 exponent conjectured by Bollobás and Erdos and matching the predictions of Gordon, Kuperberg, Patashnik, and Spencer (1996). For the upper bound, for any fixed ε>0 we construct a family of exp(O(1/ε)) graphs by gluing O(1/ε) triangles sequentially in a prescribed manner, and dynamically track the number of all homomorphisms from them, rooted at any two vertices, up to the point where n<sup>3/2+ε</sup> 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<sup>3/2-o(1)</sup> edges left.

Original languageEnglish (US)
Pages (from-to)379-438
Number of pages60
JournalAdvances in Mathematics
Volume280
DOIs
StatePublished - Aug 6 2015

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Triangle
Erdös
Terminate
Upper bound
Lower bound
Triangle-free Graph
Gluing
Greedy Algorithm
Homomorphisms
Random Graphs
Martingale
Complete Graph
Estimate
Stochastic Processes
Disjoint
Random variable
Exponent
Prediction
Graph in graph theory

Keywords

  • Random graph processes
  • Random graphs
  • Self-correcting stochastic processes
  • Triangle free graphs
  • Triangle packing

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Random triangle removal. / Bohman, Tom; Frieze, Alan; Lubetzky, Eyal.

In: Advances in Mathematics, Vol. 280, 06.08.2015, p. 379-438.

Research output: Contribution to journalArticle

Bohman, Tom ; Frieze, Alan ; Lubetzky, Eyal. / Random triangle removal. In: Advances in Mathematics. 2015 ; Vol. 280. pp. 379-438.
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