### 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 language | English (US) |
---|---|

Pages (from-to) | 379-438 |

Number of pages | 60 |

Journal | Advances in Mathematics |

Volume | 280 |

DOIs | |

State | Published - Aug 6 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*280*, 379-438. https://doi.org/10.1016/j.aim.2015.04.015

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 280, pp. 379-438. https://doi.org/10.1016/j.aim.2015.04.015

}

TY - JOUR

T1 - Random triangle removal

AU - Bohman, Tom

AU - Frieze, Alan

AU - Lubetzky, Eyal

PY - 2015/8/6

Y1 - 2015/8/6

N2 - 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 n3/2. An upper bound of o(n2) 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 n7/4+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 n3/2+o(1), 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 n3/2+ε 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 n3/2-o(1) edges left.

AB - 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 n3/2. An upper bound of o(n2) 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 n7/4+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 n3/2+o(1), 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 n3/2+ε 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 n3/2-o(1) edges left.

KW - Random graph processes

KW - Random graphs

KW - Self-correcting stochastic processes

KW - Triangle free graphs

KW - Triangle packing

UR - http://www.scopus.com/inward/record.url?scp=84928882874&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928882874&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2015.04.015

DO - 10.1016/j.aim.2015.04.015

M3 - Article

VL - 280

SP - 379

EP - 438

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -