247,746 research outputs found
Monochromatic connected matchings in 2-edge-colored multipartite graphs
A matching in a graph is connected if all the edges of are in the
same component of . Following \L uczak,there have been many results using
the existence of large connected matchings in cluster graphs with respect to
regular partitions of large graphs to show the existence of long paths and
other structures in these graphs. We prove exact
Ramsey-type bounds on the sizes of monochromatic connected matchings in
-edge-colored multipartite graphs. In addition, we prove a stability theorem
for such matchings.Comment: 29 pages, 2 figure
Continuum Percolation in the Intrinsically Secure Communications Graph
The intrinsically secure communications graph (iS-graph) is a random graph
which captures the connections that can be securely established over a
large-scale network, in the presence of eavesdroppers. It is based on
principles of information-theoretic security, widely accepted as the strictest
notion of security. In this paper, we are interested in characterizing the
global properties of the iS-graph in terms of percolation on the infinite
plane. We prove the existence of a phase transition in the Poisson iS-graph,
whereby an unbounded component of securely connected nodes suddenly arises as
we increase the density of legitimate nodes. Our work shows that long-range
communication in a wireless network is still possible when a secrecy constraint
is present.Comment: Accepted in the IEEE International Symposium on Information Theory
and its Applications (ISITA'10), Taichung, Taiwan, Oct. 201
The edge slide graph of the n-dimensional cube : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand
The goal of this thesis is to understand the spanning trees of the n-dimensional cube Qn by
understanding their edge slide graph. An edge slide is a move that “slides” an edge of a spanning
tree of Qn across a two-dimensional face, and the edge slide graph is the graph on the spanning
trees of Qn with an edge between two trees if they are connected by an edge slide. Edge slides
are a restricted form of an edge move, in which the edges involved in the move are constrained
by the structure of Qn, and the edge slide graph is a subgraph of the tree graph of Qn given by
edge moves.
The signature of a spanning tree of Qn is the n-tuple (a1; : : : ; an), where ai is the number of
edges in the ith direction. The signature of a tree is invariant under edge slides and is therefore
constant on connected components. We say that a signature is connected if the trees with that
signature lie in a single connected component, and disconnected otherwise. The goal of this
research is to determine which signatures are connected.
Signatures can be naturally classified as reducible or irreducible, with the reducible signatures
being further divided into strictly reducible and quasi-irreducible signatures. We determine
necessary and sufficient conditions for (a1; : : : ; an) to be a signature of Qn, and show that
strictly reducible signatures are disconnected. We conjecture that strict reducibility is the only
obstruction to connectivity, and present substantial partial progress towards an inductive proof
of this conjecture. In particular, we reduce the inductive step to the problem of proving under
the inductive hypothesis that every irreducible signature has a “splitting signature” for which
the upright trees with that signature and splitting signature all lie in the same component. We
establish this step for certain classes of signatures, but at present are unable to complete it for
all.
Hall’s Theorem plays an important role throughout the work, both in characterising the
signatures, and in proving the existence of certain trees used in the arguments
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Limiting shape for first-passage percolation models on random geometric graphs
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times
Percolation and Connectivity in the Intrinsically Secure Communications Graph
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 aims to characterize the global
properties of the iS-graph in terms of: (i) percolation on the infinite plane,
and (ii) full connectivity on a finite region. First, for the Poisson iS-graph
defined on the infinite plane, the existence of a phase transition is proven,
whereby an unbounded component of connected nodes suddenly arises as the
density of legitimate nodes is increased. This shows that long-range secure
communication is still possible in the presence of eavesdroppers. Second, full
connectivity on a finite region of the Poisson iS-graph is considered. The
exact asymptotic behavior of full connectivity in the limit of a large density
of legitimate nodes is characterized. Then, simple, explicit expressions are
derived in order to closely approximate the probability of full connectivity
for a finite density of legitimate nodes. The results help clarify how the
presence of eavesdroppers can compromise long-range secure communication.Comment: Submitted for journal publicatio
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
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