57,823 research outputs found
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
Minimizing the number of independent sets in triangle-free regular graphs
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the
result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the
independence polynomial of a -regular graph is maximized by disjoint copies
of . Their proof uses linear programming bounds on the distribution of
a cleverly chosen random variable. In this paper, we use this method to give
lower bounds on the independence polynomial of regular graphs. We also give new
bounds on the number of independent sets in triangle-free regular graphs
Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
Many natural and social systems develop complex networks, that are usually
modelled as random graphs. The eigenvalue spectrum of these graphs provides
information about their structural properties. While the semi-circle law is
known to describe the spectral density of uncorrelated random graphs, much less
is known about the eigenvalues of real-world graphs, describing such complex
systems as the Internet, metabolic pathways, networks of power stations,
scientific collaborations or movie actors, which are inherently correlated and
usually very sparse. An important limitation in addressing the spectra of these
systems is that the numerical determination of the spectra for systems with
more than a few thousand nodes is prohibitively time and memory consuming.
Making use of recent advances in algorithms for spectral characterization, here
we develop new methods to determine the eigenvalues of networks comparable in
size to real systems, obtaining several surprising results on the spectra of
adjacency matrices corresponding to models of real-world graphs. We find that
when the number of links grows as the number of nodes, the spectral density of
uncorrelated random graphs does not converge to the semi-circle law.
Furthermore, the spectral densities of real-world graphs have specific features
depending on the details of the corresponding models. In particular, scale-free
graphs develop a triangle-like spectral density with a power law tail, while
small-world graphs have a complex spectral density function consisting of
several sharp peaks. These and further results indicate that the spectra of
correlated graphs represent a practical tool for graph classification and can
provide useful insight into the relevant structural properties of real
networks.Comment: 14 pages, 9 figures (corrected typos, added references) accepted for
Phys. Rev.
On the average size of independent sets in triangle-free graphs
We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on vertices with maximum degree , showing that an independent set drawn uniformly at random from such a graph has expected size at least . This gives an alternative proof of Shearer's upper bound on the Ramsey number . We then prove that the total number of independent sets in a triangle-free graph with maximum degree is at least . The constant in the exponent is best possible. In both cases, tightness is exhibited by a random -regular graph. Both results come from considering the hard-core model from statistical physics: a random independent set drawn from a graph with probability proportional to , for a fugacity parameter . We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree . The bound is asymptotically tight in for all . We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theor
On existentially complete triangle-free graphs
For a positive integer k, we say that a graph is k-existentially complete if for every 0 ⩽ a ⩽ k, and every tuple of distinct vertices x1, …, xa, y1, …, yk−a, there exists a vertex z that is joined to all of the vertices x1, …, xa and to none of the vertices y1, …, yk−a. While it is easy to show that the binomial random graph Gn,1/2 satisfies this property (with high probability) for k = (1 − o(1)) log2n, little is known about the “triangle-free” version of this problem: does there exist a finite triangle-free graph G with a similar “extension property”? This question was first raised by Cherlin in 1993 and remains open even in the case k = 4.
We show that there are no k-existentially complete triangle-free graphs on n vertices with k>8lognloglogn, for n sufficiently large
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
Inapproximability for Antiferromagnetic Spin Systems in the Tree Non-Uniqueness Region
A remarkable connection has been established for antiferromagnetic 2-spin
systems, including the Ising and hard-core models, showing that the
computational complexity of approximating the partition function for graphs
with maximum degree D undergoes a phase transition that coincides with the
statistical physics uniqueness/non-uniqueness phase transition on the infinite
D-regular tree. Despite this clear picture for 2-spin systems, there is little
known for multi-spin systems. We present the first analog of the above
inapproximability results for multi-spin systems.
The main difficulty in previous inapproximability results was analyzing the
behavior of the model on random D-regular bipartite graphs, which served as the
gadget in the reduction. To this end one needs to understand the moments of the
partition function. Our key contribution is connecting: (i) induced matrix
norms, (ii) maxima of the expectation of the partition function, and (iii)
attractive fixed points of the associated tree recursions (belief propagation).
The view through matrix norms allows a simple and generic analysis of the
second moment for any spin system on random D-regular bipartite graphs. This
yields concentration results for any spin system in which one can analyze the
maxima of the first moment. The connection to fixed points of the tree
recursions enables an analysis of the maxima of the first moment for specific
models of interest.
For k-colorings we prove that for even k, in the tree non-uniqueness region
(which corresponds to k<D) it is NP-hard, unless NP=RP, to approximate the
number of colorings for triangle-free D-regular graphs. Our proof extends to
the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic
model under a mild condition
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