135 research outputs found
Random triangle removal
Starting from a complete graph on 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 , motivated by the study of the Ramsey
number . An upper bound of 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 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 , 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 we construct a family of
graphs by gluing triangles sequentially
in a prescribed manner, and dynamically track all homomorphisms from them,
rooted at any two vertices, up to the point where 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 edges
left.Comment: 42 pages, 4 figures. Supercedes arXiv:1108.178
Explicit expanders with cutoff phenomena
The cutoff phenomenon describes a sharp transition in the convergence of an
ergodic finite Markov chain to equilibrium. Of particular interest is
understanding this convergence for the simple random walk on a bounded-degree
expander graph. The first example of a family of bounded-degree graphs where
the random walk exhibits cutoff in total-variation was provided only very
recently, when the authors showed this for a typical random regular graph.
However, no example was known for an explicit (deterministic) family of
expanders with this phenomenon. Here we construct a family of cubic expanders
where the random walk from a worst case initial position exhibits
total-variation cutoff. Variants of this construction give cubic expanders
without cutoff, as well as cubic graphs with cutoff at any prescribed
time-point.Comment: 17 pages, 2 figure
- …