Combinatorics Seminar

When: Sunday, January 12, 10am
Where: Schreiber 309
Speaker: Eyal Lubetzky, Microsoft Research Redmond
Title: Random Triangle Removal


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). Bollobas and Erdos (1990) conjectured that the expected final number of edges has order n^{3/2}. An upper bound of o(n^2) was shown by Spencer (1995) and independently by Rodl 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 Bollobas and Erdos 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 Bollobas and Erdos 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 vertex, 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.

Joint work with Tom Bohman and Alan Frieze.