Extremal Graph Theory

Extremal Graph Theory

This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory.

Author: Bela Bollobas

Publisher: Courier Corporation

ISBN: 9780486317588

Category: Mathematics

Page: 512

View: 698

The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory. Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics. Pure mathematicians will find this text a valuable resource in terms of its unusually large collection of results and proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.
Categories: Mathematics

Extremal Graph Theory with Emphasis on Probabilistic Methods

Extremal Graph Theory with Emphasis on Probabilistic Methods

In this book, an update of his 1978 book Extremal Graph Theory, the author focuses on a trend towards probabilistic methods.

Author: Béla Bollobás

Publisher: American Mathematical Soc.

ISBN: 9780821807125

Category: Mathematics

Page: 64

View: 624

Problems in extremal graph theory have traditionally been tackled by ingenious methods which made use of the structure of extremal graphs. In this book, an update of his 1978 book ""Extremal Graph Theory"", the author focuses on a trend towards probabilistic methods. He demonstrates both the direct use of probability theory and, more importantly, the fruitful adoption of a probabilistic frame of mind when tackling main line extremal problems. Essentially self-contained, the book does not merely catalog results, but rather includes considerable discussion on a few of the deeper results. The author addresses pure mathematicians, especially combinatorialists and graduate students taking graph theory, as well as theoretical computer scientists. He assumes a mature familiarity with combinatorial methods and an acquaintance with basic graph theory. The book is based on the NSF-CBMS Regional Conference on Graph Theory held at Emory University in June, 1984.
Categories: Mathematics

Some Tur n type Problems in Extremal Graph Theory

Some Tur  n type Problems in Extremal Graph Theory

Since the seminal work of Turán, the forbidden subgraph problem has been among the central questions in extremal graph theory.

Author: Jangwon Yie

Publisher:

ISBN: OCLC:1156472092

Category: Extremal problems (Mathematics)

Page: 98

View: 790

Since the seminal work of Turán, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n; F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F) for various graphs F. The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Turán-type problem. In this thesis, we will study Turán-type problems and their variants for graphs and hypergraphs. Chapter 2 contains a Turán-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs. In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures. Chapter 4 includes another Turán-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
Categories: Extremal problems (Mathematics)

Extremal Graph Theory with Emphasis on Probabilistic Methods

Extremal Graph Theory with Emphasis on Probabilistic Methods

In this book, an update of his 1978 book Extremal Graph Theory, the author focuses on a trend towards probabilistic methods.

Author: Béla Bollobás

Publisher: American Mathematical Soc.

ISBN: 0821889079

Category: Mathematics

Page: 64

View: 771

Problems in extremal graph theory have traditionally been tackled by ingenious methods which made use of the structure of extremal graphs. In this book, an update of his 1978 book Extremal Graph Theory, the author focuses on a trend towards probabilistic methods. He demonstrates both the direct use of probability theory and, more importantly, the fruitful adoption of a probabilistic frame of mind when tackling main line extremal problems. Essentially self-contained, the book doesnot merely catalog results, but rather includes considerable discussion on a few of the deeper results. The author addresses pure mathematicians, especially combinatorialists and graduate students taking graph theory, as well as theoretical computer scientists. He assumes a mature familiarity withcombinatorial methods and an acquaintance with basic graph theory. The book is based on the NSF-CBMS Regional Conference on Graph Theory held at Emory University in June, 1984.
Categories: Mathematics

Three Existence Problems in Extremal Graph Theory

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.

Author: Paul S. Wenger

Publisher:

ISBN: OCLC:774920501

Category:

Page:

View: 512

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 $G$ is an assignment of intervals on the real line to the vertices of $G$ such that two vertices are adjacent if and only if their intervals intersect. We consider graphs that have {it bar $k$-visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations. We obtain results on $mathcal{F}_k$, the family of graphs having bar $k$-visibility representations. We also study $bigcup_{k=0}^{infty} mathcal{F}_k$. In particular, we determine the largest complete graph having a bar $k$-visibility representation, and we show that there are graphs that do not have bar $k$-visibility representations for any $k$. 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 $G$ 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 $G$ on the minimum number of vertices; this minimum number is the {it acquisition number} of $G$. 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 $2$. 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 $H$, we consider those graphs that do not contain $H$ and such that the addition of any edge completes exactly one copy of $H$. Such a graph is called {it uniquely $H$-saturated}. We study the existence of uniquely $H$-saturated graphs when $H$ is a path or a cycle. In particular, we determine all of the uniquely $C_4$-saturated graphs; there are exactly ten. Interestingly, the uniquely $C_{5}$-saturated graphs are precisely the friendship graphs characterized by Wilf.
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Topics in Extremal Graph Theory

Topics in Extremal Graph Theory

Author: Brandon Cruz

Publisher:

ISBN: OCLC:551177066

Category: Extremal problems (Mathematics)

Page: 120

View: 259

Categories: Extremal problems (Mathematics)

Extremal Graph Theory

Extremal Graph Theory

John Dryden 's “The Spanish Friar” Extremal graph theory, in its strictest sense, is
a branch of graph theory developed and loved by Hungarians. Its study, as a
subject in its own right, was initiated by Turétn in 1940, although a special case of
 ...

Author: Béla Bollobás

Publisher: Courier Corporation

ISBN: 9780486435961

Category: Mathematics

Page: 488

View: 289

The ever-expanding field of extremal graph theory encompasses an array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume presents a concise yet comprehensive treatment, featuring complete proofs for almost all of its results and numerous exercises. 1978 edition.
Categories: Mathematics

Extremal Graph Theory Ramsey Tur n Numbers Chromatic Thresholds and Minors

Extremal Graph Theory  Ramsey Tur  n Numbers  Chromatic Thresholds  and Minors

This dissertation investigates several questions in extremal graph theory and the theory of graph minors.

Author: John E. Lenz

Publisher:

ISBN: OCLC:774893660

Category:

Page:

View: 691

This dissertation investigates several questions in extremal graph theory and the theory of graph minors. It consists of three independent parts; the first two parts focus on questions motivated by Turan's Theorem and the third part investigates a problem related to Hadwiger's Conjecture. Let H be a graph, t an integer, and f(n) a function. The t-Ramsey-Turan number of H, RT_t(n,H,f(n)), is the maximum number of edges in an n-vertex, H-free graph with K_t-independence number less than f(n), where the K_t-independence number of a graph G is the maximum number of vertices in a K_t-free induced graph of G. In the first part of this thesis, we study the Ramsey-Turan numbers for several graphs and hypergraphs, proving two conjectures of Erdos, Hajnal, Simonovits, Sos, and Szemeredi. In joint work with Jozsef Balogh, our first main theorem is to provide the first lower bounds of order Omega(n^2) on RT_t(n,K_{t+2},o(n)). Our second main theorem is to prove lower bounds on RT(n,tk{r}{s},o(n)), where tk{r}{s} is the r-uniform hypergraph formed from K_s by adding r-2 new vertices to every edge. Let mathcal{F} be a family of r-uniform hypergraphs. Introduced by Erdos and Simonovits, the chromatic threshold of mathcal{F} is the infimum of the values c >= 0 such that the subfamily of mathcal{F} consisting of hypergraphs with minimum degree at least $cbinom{n}{r-1}$ has bounded chromatic number. The problem of chromatic thresholds of graphs has been well studied, but there have been no previous results about the chromatic thresholds of r-uniform hypergraphs for r >= 3. Our main result in this part of the thesis, in joint work with Jozsef Balogh, Jane Butterfield, Ping Hu, and Dhruv Mubayi, is to prove a structural theorem about hypergraphs with bounded chromatic number. Corollaries of this result show that the chromatic threshold of the family of F-free hypergraphs is zero for several hypergraphs F, including a hypergraph generalization of cycles. A graph H is a minor of a graph G if starting with G, one can obtain H by a sequence of vertex deletions, edge deletions, and edge contractions. Hadwiger's famous conjecture from 1943 states that every t-chromatic graph G has K_t as a minor. Hadwiger's Conjecture implies the following weaker conjecture: every graph G has $K_{leftlceil n/alpha(G) rightrceil}$ as a minor, where alpha(G) is the independence number of G. The main theorem in the last part of this thesis, in joint work with Jozsef Balogh and Hehui Wu, is to prove that every graph has $K_{n/(2alpha(G) - Theta(log alpha(G)))}$ as a minor.
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