25,503 research outputs found
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
On two problems in Ramsey-Tur\'an theory
Alon, Balogh, Keevash and Sudakov proved that the -partite Tur\'an
graph maximizes the number of distinct -edge-colorings with no monochromatic
for all fixed and , among all -vertex graphs. In this
paper, we determine this function asymptotically for among -vertex
graphs with sub-linear independence number. Somewhat surprisingly, unlike
Alon-Balogh-Keevash-Sudakov's result, the extremal construction from
Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of
distinct edge-colorings with no monochromatic cliques among all graphs with
sub-linear independence number, even in the 2-colored case.
In the second problem, we determine the maximum number of triangles
asymptotically in an -vertex -free graph with . The
extremal graphs have similar structure to the extremal graphs for the classical
Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
We consider sequences of graphs and define various notions of convergence
related to these sequences: ``left convergence'' defined in terms of the
densities of homomorphisms from small graphs into the graphs of the sequence,
and ``right convergence'' defined in terms of the densities of homomorphisms
from the graphs of the sequence into small graphs; and convergence in a
suitably defined metric.
In Part I of this series, we show that left convergence is equivalent to
convergence in metric, both for simple graphs, and for graphs with nodeweights
and edgeweights. One of the main steps here is the introduction of a
cut-distance comparing graphs, not necessarily of the same size. We also show
how these notions of convergence provide natural formulations of Szemeredi
partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This
version differs from an earlier version from May 2006 in the organization of
the sections, but is otherwise almost identica
Two conjectures in Ramsey-Tur\'an theory
Given graphs , a graph is -free if
there is a -edge-colouring with no monochromatic
copy of with edges of colour for each . Fix a function
, the Ramsey-Tur\'an function is the
maximum number of edges in an -vertex -free graph with
independence number at most . We determine for and sufficiently small , confirming a
conjecture of Erd\H{o}s and S\'os from 1979. It is known that
has a phase transition at . However, the values of was not
known. We determined this value by proving , answering a question of Balogh, Hu and Simonovits.
The proofs utilise, among others, dependent random choice and results from
graph packings.Comment: 20 pages, 2 figures, 2 pages appendi
- β¦