1,051 research outputs found
Cycle lengths in sparse graphs
Let C(G) denote the set of lengths of cycles in a graph G. In the first part
of this paper, we study the minimum possible value of |C(G)| over all graphs G
of average degree d and girth g. Erdos conjectured that |C(G)|
=\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this
conjecture. In particular, the longest cycle in a graph of average degree d and
girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this
problem was initiated by Ore in 1967 and our result improves all previously
known lower bounds on the length of the longest cycle. Moreover, our bound
cannot be improved in general, since known constructions of d-regular Moore
Graphs of girth g have roughly that many vertices. We also show that
\Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle
lengths in a graph of chromatic number d and girth g. Further results are
obtained for the number of cycle lengths in H-free graphs of average degree d.
In the second part of the paper, motivated by the conjecture of Erdos and
Gyarfas that every graph of minimum degree at least three contains a cycle of
length a power of two, we prove a general theorem which gives an upper bound on
the average degree of an n-vertex graph with no cycle of even length in a
prescribed infinite sequence of integers. For many sequences, including the
powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average
degree of graph of order n with no cycle of length in the sequence, where
\log^* n is the number of times the binary logarithm must be applied to n to
get a number which is at mos
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Learning loopy graphical models with latent variables: Efficient methods and guarantees
The problem of structure estimation in graphical models with latent variables
is considered. We characterize conditions for tractable graph estimation and
develop efficient methods with provable guarantees. We consider models where
the underlying Markov graph is locally tree-like, and the model is in the
regime of correlation decay. For the special case of the Ising model, the
number of samples required for structural consistency of our method scales
as , where p is the
number of variables, is the minimum edge potential, is
the depth (i.e., distance from a hidden node to the nearest observed nodes),
and is a parameter which depends on the bounds on node and edge
potentials in the Ising model. Necessary conditions for structural consistency
under any algorithm are derived and our method nearly matches the lower bound
on sample requirements. Further, the proposed method is practical to implement
and provides flexibility to control the number of latent variables and the
cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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