111 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
Dynamic concentration of the triangle-free process
The triangle-free process begins with an empty graph on n vertices and
iteratively adds edges chosen uniformly at random subject to the constraint
that no triangle is formed. We determine the asymptotic number of edges in the
maximal triangle-free graph at which the triangle-free process terminates. We
also bound the independence number of this graph, which gives an improved lower
bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t),
which is within a 4+o(1) factor of the best known upper bound. Our improvement
on previous analyses of this process exploits the self-correcting nature of key
statistics of the process. Furthermore, we determine which bounded size
subgraphs are likely to appear in the maximal triangle-free graph produced by
the triangle-free process: they are precisely those triangle-free graphs with
density at most 2.Comment: 75 pages, 1 figur
The game chromatic number of random graphs
Given a graph G and an integer k, two players take turns coloring the
vertices of G one by one using k colors so that neighboring vertices get
different colors. The first player wins iff at the end of the game all the
vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k
for which the first player has a winning strategy. In this paper we analyze the
asymptotic behavior of this parameter for a random graph G_{n,p}. We show that
with high probability the game chromatic number of G_{n,p} is at least twice
its chromatic number but, up to a multiplicative constant, has the same order
of magnitude. We also study the game chromatic number of random bipartite
graphs
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