Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_4228
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
v, 66 p.
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Robert DeMarco
Abstract (type = abstract)
We prove four separate results. These results will appear or have appeared in various
papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader
is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξn,p
k the number
of copies of Kk in G, p ≥ n−2/(k−1) and η > 0, we show when k > 1
Pr(ξk > (1 + η)Eξk) < exp
[
−Ωη,k min{n2pk−1 log(1/p), nkp(k
2)}
]
.
This is tight up to the value of the constant in the exponent.
For a graph H, denote by t(H) (resp. b(H)) the maximum size of a trianglefree
(resp. bipartite) subgraph of H. We show that w.h.p. t(G) = b(G) if p >
Cn−1/2 log1/2 n for a suitable constant C, which is best possible up to the value of C.
We give a new (simpler) proof of a random analogue of the Erd˝os-Simonovits “stability”
version of Mantel’s Theorem, viz.: For each η > 0 there is a C such that if
p > Cn−1/2, then w.h.p. each triangle-free subgraph of G of size at least |G|/2 can be
made bipartite by deletion of at most ηn2p edges.
Let C(H) denote the cycle space and T (H) the triangle space of a graph H. We use
the previous result to show that if C >
√
3/2 is fixed and p > C
√
log n/n, then w.h.p.
T (G) = C(G). The lower bound on p is best possible.
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
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.