697,623 research outputs found
Chromatic Polynomials for Strip Graphs and their Asymptotic Limits
We calculate the chromatic polynomials for -vertex strip graphs of the
form , where and are various subgraphs on the
left and right ends of the strip, whose bulk is comprised of -fold
repetitions of a subgraph . The strips have free boundary conditions in the
longitudinal direction and free or periodic boundary conditions in the
transverse direction. This extends our earlier calculations for strip graphs of
the form . We use a generating function method. From
these results we compute the asymptotic limiting function ; for this has physical significance as
the ground state degeneracy per site (exponent of the ground state entropy) of
the -state Potts antiferromagnet on the given strip. In the complex
plane, is an analytic function except on a certain continuous locus . In contrast to the strip graphs, where
(i) is independent of , and (ii) consists of arcs and possible line segments
that do not enclose any regions in the plane, we find that for some
strip graphs, (i) does depend on and
, and (ii) can enclose regions in the plane. Our study elucidates the
effects of different end subgraphs and and of boundary conditions on
the infinite-length limit of the strip graphs.Comment: 33 pages, Latex, 7 encapsulated postscript figures, Physica A, in
press, with some typos fixe
Planting trees in graphs, and finding them back
In this paper we study detection and reconstruction of planted structures in
Erd\H{o}s-R\'enyi random graphs. Motivated by a problem of communication
security, we focus on planted structures that consist in a tree graph. For
planted line graphs, we establish the following phase diagram. In a low density
region where the average degree of the initial graph is below some
critical value , detection and reconstruction go from impossible
to easy as the line length crosses some critical value ,
where is the number of nodes in the graph. In the high density region
, detection goes from impossible to easy as goes from
to , and reconstruction remains impossible so
long as . For -ary trees of varying depth and ,
we identify a low-density region , such that the following
holds. There is a threshold with the following properties.
Detection goes from feasible to impossible as crosses . We also show
that only partial reconstruction is feasible at best for . We
conjecture a similar picture to hold for -ary trees as for lines in the
high-density region , but confirm only the following part of
this picture: Detection is easy for -ary trees of size ,
while at best only partial reconstruction is feasible for -ary trees of any
size . These results are in contrast with the corresponding picture for
detection and reconstruction of {\em low rank} planted structures, such as
dense subgraphs and block communities: We observe a discrepancy between
detection and reconstruction, the latter being impossible for a wide range of
parameters where detection is easy. This property does not hold for previously
studied low rank planted structures
Three Existence Problems in Extremal Graph Theory
Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics.
In this thesis we take this approach to three different structural questions rooted in extremal graph theory.
When studying graph representations, we seek efficient ways to encode the structure of a graph.
For example, an {\it interval representation} of a graph is an assignment of intervals on the real line to the vertices of such that two vertices are adjacent if and only if their intervals intersect.
We consider graphs that have {\it bar -visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations.
We obtain results on , the family of graphs having bar -visibility representations.
We also study .
In particular, we determine the largest complete graph having a bar -visibility representation, and we show that there are graphs that do not have bar -visibility representations for any .
Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs.
Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around provided that each move transfers weight from a vertex to a heavier neighbor.
Our goal in making acquisition moves is to consolidate all of the weight in on the minimum number of vertices; this minimum number is the {\it acquisition number} of .
We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight.
We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter .
We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex.
Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects.
Some local conditions are so limiting that very few objects satisfy the requirements.
For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor.
Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex.
We study a related structural restriction where similar phenomena occur.
For a fixed graph , we consider those graphs that do not contain and such that the addition of any edge completes exactly one copy of .
Such a graph is called {\it uniquely -saturated}.
We study the existence of uniquely -saturated graphs when is a path or a cycle.
In particular, we determine all of the uniquely -saturated graphs; there are exactly ten.
Interestingly, the uniquely -saturated graphs are precisely the friendship graphs characterized by Wilf
The Infinite Server Problem
We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which implies the same lower bound for the (h,k)-server problem even when k>>h and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case
Complexity of Computing the Anti-Ramsey Numbers for Paths
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced
in 1978, by Erd\" os, Simonovits and S\' os. For given graphs and the
\emph{anti-Ramsey number} is defined to be the maximum
number such that there exists an assignment of colors to the edges of
in which every copy of in has at least two edges with the same
color.
There are works on the computational complexity of the problem when is a
star. Along this line of research, we study the complexity of computing the
anti-Ramsey number , where is a path of length .
First, we observe that when , the problem is hard; hence, the
challenging part is the computational complexity of the problem when is a
fixed constant.
We provide a characterization of the problem for paths of constant length.
Our first main contribution is to prove that computing for
every integer is NP-hard. We obtain this by providing several structural
properties of such coloring in graphs. We investigate further and show that
approximating to a factor of is hard
already in -partite graphs, unless P=NP. We also study the exact complexity
of the precolored version and show that there is no subexponential algorithm
for the problem unless ETH fails for any fixed constant .
Given the hardness of approximation and parametrization of the problem, it is
natural to study the problem on restricted graph families. We introduce the
notion of color connected coloring and employing this structural property. We
obtain a linear time algorithm to compute , for every
integer , when the host graph, , is a tree
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