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Embedding spanning subgraphs into large dense graphs

## Descriptive

TypeOfResource
Text
TitleInfo (ID = T-1)
Title
Embedding spanning subgraphs into large dense graphs
Identifier
ETD_2800
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000056417
Language
LanguageTerm (authority = ISO639-2); (type = code)
eng
Genre (authority = marcgt)
theses
Subject (ID = SBJ-1); (authority = RUETD)
Topic
Computer Science
Subject (ID = SBJ-2); (authority = ETD-LCSH)
Topic
Hamiltonian graph theory
Subject (ID = SBJ-3); (authority = ETD-LCSH)
Topic
Spanning trees (Graph theory)
Subject (ID = SBJ-4); (authority = ETD-LCSH)
Topic
Embeddings (Mathematics)
Abstract
In this thesis we are going to present some results on embedding spanning subgraphs into large dense graphs. Spanning Trees Bollob'as conjectured that if \$G\$ is a graph on \$n\$ vertices, \$delta(G) geq (1/2 + epsilon) n\$ for some \$epsilon > 0\$, and \$T\$ is a bounded degree tree on \$n\$ vertices, then \$T\$ is a subgraph of \$G\$. The problem was solved in the affirmative by Koml'os, S'ark"ozy and Szemer'edi for large graphs. They then strengthened their result, and showed that the maximum degree of \$T\$ need not be bounded: there exists a constant \$c\$ such that \$T\$ is a subgraph of \$G\$ if \$Delta(T) leq cn / log n\$, \$delta(G) geq (1/2 + epsilon) n\$ and \$n\$ is large. Both proofs are based on the Regularity Lemma-Blow-up Lemma Method. Recently, using other methods, it was shown that bounded degree trees embed into graphs with minimum degree \$n/2 + C log n\$, where \$C\$ is a constant depending on the maximum degree of \$T\$. Here we show that in general \$n/2 + O(Delta(T) cdot log n)\$ is sufficient for every \$Delta(T) leq cn / log n\$. We also show that this bound is tight for the two extreme values of \$m\$ i.e. when \$m = C\$ and when \$m = cn / log n\$. Powers of Hamiltonian Cycles In 1962 P'osa conjectured that if \$delta(G) geq frac{2}{3}n\$ then \$G\$ contains the square of a Hamiltonian cycle. Later, in 1974, Seymour generalized this conjecture: if \$delta(G) geq (frac{k-1}{k})n\$ then \$G\$ contains the \$(k-1)\$th power of a Hamiltonian cycle. In 1998 the conjecture was proved by Koml'os, S'ark"ozy and Szemer'edi for large graphs using the Regularity Lemma. We present a ``deregularised" proof of the P'osa-Seymour conjecture which results in a much lower threshold value for \$n\$, the size of the graph for which the conjecture is true. We hope that the tools used in this proof will push down the threshold value for \$n\$ to around 100 at which point we will be able to verify the conjecture for every \$n\$.
PhysicalDescription
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electronic resource
Extent
xi, 95 p. : ill.
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application/pdf
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text/xml
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = vita)
Includes vita
Note (type = statement of responsibility)
by Asif Jamshed
Name (ID = NAME-1); (type = personal)
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Jamshed
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Asif
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author
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Asif Jamshed
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Szemeredi
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Endre
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chair
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Endre Szemeredi
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Steiger
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William
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internal member
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William Steiger
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Michael
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internal member
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Abello
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James
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James Abello
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Rutgers University
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degree grantor
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Role
RoleTerm (authority = RULIB)
school
OriginInfo
DateCreated (qualifier = exact)
2010
DateOther (qualifier = exact); (type = degree)
2010-10
Place
PlaceTerm (type = code)
xx
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TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3MC8ZSD
Genre (authority = ExL-Esploro)
ETD doctoral
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## Rights

RightsDeclaration (AUTHORITY = GS); (ID = rulibRdec0006)
The author owns the copyright to this work.
Status
Availability
Status
Open
Reason
RightsHolder (ID = PRH-1); (type = personal)
Name
FamilyName
Jamshed
GivenName
Asif
Role
RightsEvent (ID = RE-1); (AUTHORITY = rulib)
Type
DateTime
2010-08-04 19:06:06
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Role
Name
Asif Jamshed
Affiliation
Rutgers University. Graduate School - New Brunswick
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Type
Name
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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## Technical

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ETD
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