17,070 research outputs found
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs
We establish relations between the bandwidth and the treewidth of bounded
degree graphs G, and relate these parameters to the size of a separator of G as
well as the size of an expanding subgraph of G. Our results imply that if one
of these parameters is sublinear in the number of vertices of G then so are all
the others. This implies for example that graphs of fixed genus have sublinear
bandwidth or, more generally, a corresponding result for graphs with any fixed
forbidden minor. As a consequence we establish a simple criterion for
universality for such classes of graphs and show for example that for each
gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy
of every bounded-degree planar graph on n vertices if n is sufficiently large
A Local Algorithm for the Sparse Spanning Graph Problem
Constructing a sparse spanning subgraph is a fundamental primitive in graph
theory. In this paper, we study this problem in the Centralized Local model,
where the goal is to decide whether an edge is part of the spanning subgraph by
examining only a small part of the input; yet, answers must be globally
consistent and independent of prior queries.
Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees,
cannot be constructed efficiently in this model. Therefore, we settle for a
spanning subgraph containing at most edges (where is the
number of vertices and is a given approximation/sparsity
parameter). We achieve query complexity of
, (-notation hides
polylogarithmic factors in ). where is the maximum degree of the
input graph. Our algorithm is the first to do so on arbitrary bounded degree
graphs. Moreover, we achieve the additional property that our algorithm outputs
a spanner, i.e., distances are approximately preserved. With high probability,
for each deleted edge there is a path of
hops in the output that connects its endpoints
Embedding nearly-spanning bounded degree trees
We derive a sufficient condition for a sparse graph G on n vertices to
contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n
vertices, in terms of the expansion properties of G. As a result we show that
for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon)
such that a random graph G(n,c/n) contains almost surely a copy of every tree T
on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if
an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose
eigenvalues, except the first one, are at most \lambda in their absolute
values) has large enough spectral gap D/\lambda as a function of d and
\epsilon, then G has a copy of every tree T as above
Ising critical exponents on random trees and graphs
We study the critical behavior of the ferromagnetic Ising model on random
trees as well as so-called locally tree-like random graphs. We pay special
attention to trees and graphs with a power-law offspring or degree distribution
whose tail behavior is characterized by its power-law exponent . We
show that the critical temperature of the Ising model equals the inverse
hyperbolic tangent of the inverse of the mean offspring or mean forward degree
distribution. In particular, the inverse critical temperature equals zero when
where this mean equals infinity.
We further study the critical exponents and ,
describing how the (root) magnetization behaves close to criticality. We
rigorously identify these critical exponents and show that they take the values
as predicted by Dorogovstev, et al. and Leone et al. These values depend on the
power-law exponent , taking the mean-field values for , but
different values for
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