4,438 research outputs found
Subdivisions with Distance Constraints in Large Graphs
In this dissertation we are concerned with sharp degree conditions that guarantee the existence of certain types of subdivisions in large graphs. Of particular interest are subdivisions with a certain number of arbitrarily specified vertices and with prescribed path lengths. Our non-standard approach makes heavy use of the Regularity Lemma (Szemerédi, 1978), the Blow-Up Lemma (Komlós, Sárkózy, and Szemerédi, 1994), and the minimum degree panconnectivity criterion (Williamson, 1977).Sharp minimum degree criteria for a graph G to be H-linked have recently been discovered. We define (H,w,d)-linkage, a condition stronger than H-linkage, by including a weighting function w consisting of required lengths for each edge-path of a desired H-subdivision. We establish sharp minimum degree criteria for a large graph G to be (H,w,d)-linked for all nonnegative d. We similarly define the weaker condition (H,S,w,d)-semi-linkage, where S denotes the set of vertices of H whose corresponding vertices in an H-subdivision are arbitrarily specified. We prove similar sharp minimum degree criteria for a large graph G to be (H,S,w,d)-semi-linked for all nonnegativeWe also examine path coverings in large graphs, which could be seen as a special case of (H,S,w)-semi-linkage. In 2000, Enomoto and Ota conjectured that a graph G of order n with degree sum σ2(G) satisfying σ2(G) \u3e n + k - 2 may be partitioned into k paths, each of prescribed order and with a specified starting vertex. We prove the Enomoto-Ota Conjecture for graphs of sufficiently large order
Precise Partitions Of Large Graphs
First by using an easy application of the Regularity Lemma, we extend some known results about cycles of many lengths to include a specified edge on the cycles. The results in this chapter will help us in rest of this thesis. In 2000, Enomoto and Ota posed a conjecture on the existence of path decomposition of graphs with fixed start vertices and fixed lengths. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices. Furthermore, sharp minimum degree and degree sum conditions are proven for the existance of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs. Finally, we prove a sharp connectivity and degree sum condition for the existence of a subdivision of a multigraph in which some of the vertices are specified and the distance between each pair of vertices in the subdivision is prescribed (within one)
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
The directed 2-linkage problem with length constraints
Postponed access: the file will be available after 2022-01-15acceptedVersio
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with vertices. We show that, modulo planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order of , and . For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety over the complex numbers . In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with hyperplanes yields at most zero-dimensional components, and one finds this degree to be . The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of for the number of spatial embeddings with generic edge lengths of the -skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the non-Euclidean case
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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