130,149 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
Field theoretic approach to metastability in the contact process
A quantum field theoretic formulation of the dynamics of the Contact Process
on a regular graph of degree z is introduced. A perturbative calculation in
powers of 1/z of the effective potential for the density of particles phi(t)
and an instantonic field psi(t) emerging from the quantum formalism is
performed. Corrections to the mean-field distribution of densities of particles
in the out-of-equilibrium stationary state are derived in powers of 1/z.
Results for typical (e.g. average density) and rare fluctuation (e.g. lifetime
of the metastable state) properties are in very good agreement with numerical
simulations carried out on D-dimensional hypercubic (z=2D) and Cayley lattices.Comment: Final published version; 20 pages, 5 figure
A solution to Erd\H{o}s and Hajnal's odd cycle problem
In 1981, Erd\H{o}s and Hajnal asked whether the sum of the reciprocals of the
odd cycle lengths in a graph with infinite chromatic number is necessarily
infinite. Let be the set of cycle lengths in a graph and
let be the set of odd numbers in .
We prove that, if has chromatic number , then . This solves Erd\H{o}s
and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically
optimal.
In 1984, Erd\H{o}s asked whether there is some such that each graph with
chromatic number at least (or perhaps even only average degree at least
) has a cycle whose length is a power of 2. We show that an average degree
condition is sufficient for this problem, solving it with methods that apply to
a wide range of sequences in addition to the powers of 2.
Finally, we use our methods to show that, for every , there is some so
that every graph with average degree at least has a subdivision of the
complete graph in which each edge is subdivided the same number of times.
This confirms a conjecture of Thomassen from 1984.Comment: 42 pages, 3 figures. Version accepted for publicatio
A solution to Erdős and Hajnal’s odd cycle problem
In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let C(G) be the set of cycle lengths in a graph G and let Codd(G) be the set of odd numbers in C(G). We prove that, if G has chromatic number k, then ∑ℓ∈Codd(G)1/ℓ≥(1/2−ok(1))logk. This solves Erdős and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal.
In 1984, Erdős asked whether there is some d such that each graph with chromatic number at least d (or perhaps even only average degree at least d) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2.
Finally, we use our methods to show that, for every k, there is some d so that every graph with average degree at least d has a subdivision of the complete graph Kk in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984
Wireless Secrecy in Large-Scale Networks
The ability to exchange secret information is critical to many commercial,
governmental, and military networks. The intrinsically secure communications
graph (iS-graph) is a random graph which describes the connections that can be
securely established over a large-scale network, by exploiting the physical
properties of the wireless medium. This paper provides an overview of the main
properties of this new class of random graphs. We first analyze the local
properties of the iS-graph, namely the degree distributions and their
dependence on fading, target secrecy rate, and eavesdropper collusion. To
mitigate the effect of the eavesdroppers, we propose two techniques that
improve secure connectivity. Then, we analyze the global properties of the
iS-graph, namely percolation on the infinite plane, and full connectivity on a
finite region. These results help clarify how the presence of eavesdroppers can
compromise secure communication in a large-scale network.Comment: To appear: Proc. IEEE Information Theory and Applications Workshop
(ITA'11), San Diego, CA, Feb. 2011, pp. 1-10, Invited Pape
Coloring and constructing (hyper)graphs with restrictions
We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties.
In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph.
In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős).
In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor.
In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases.
In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound
Maximal-entropy random walk unifies centrality measures
In this paper analogies between different (dis)similarity matrices are
derived. These matrices, which are connected to path enumeration and random
walks, are used in community detection methods or in computation of centrality
measures for complex networks. The focus is on a number of known centrality
measures, which inherit the connections established for similarity matrices.
These measures are based on the principal eigenvector of the adjacency matrix,
path enumeration, as well as on the stationary state, stochastic matrix or mean
first-passage times of a random walk. Particular attention is paid to the
maximal-entropy random walk, which serves as a very distinct alternative to the
ordinary random walk used in network analysis.
The various importance measures, defined both with the use of ordinary random
walk and the maximal-entropy random walk, are compared numerically on a set of
benchmark graphs. It is shown that groups of centrality measures defined with
the two random walks cluster into two separate families. In particular, the
group of centralities for the maximal-entropy random walk, connected to the
eigenvector centrality and path enumeration, is strongly distinct from all the
other measures and produces largely equivalent results.Comment: 7 pages, 2 figure
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