[Combprob] Thursday, Dec 3, Eyal Lubetzky on "Random Triangle Removal"

[Daniel Spielman]

Combinatorics and Probability seminar
Thursday Dec 3, 4-5pm, LOM 215.

Speaker: E. Lubetzky (NYU).
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TITLE:                    Random Triangle Removal


ABSTRACT:

Starting from a complete graph on [image: 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 Erdős (1990)
conjectured that the expected final number of edges has order [image:
n^{3/2}]. An upper bound of [image: o(n^2)] 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 Erdős was [image: 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 [image: n^{3/2+o(1)}], thus confirming
the [image: 3/2] exponent conjectured by Bollobás and Erdős and matching
the predictions of Spencer et al. For the upper bound, for any fixed [image:
\epsilon>0] we construct a family of [image: \exp(O(1/\epsilon))] graphs by
gluing [image: 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 [image: 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 [image:
n^{3/2-o(1)}] edges left.


Joint work with Tom Bohman and Alan Frieze.
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