Some Extremal Problems for Graphs with Fixed Numbers of Vertices and Edges PDF Download

Author: Felix G. Lazebnik
Publisher:
ISBN:
Category :
Languages : en
Pages : 212

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Author: Daniel Pritikin
Publisher:
ISBN:
Category : Extremal problems (Mathematics)
Languages : en
Pages : 200

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Author: Craig Weidert
Publisher:
ISBN:
Category : Extremal problems (Mathematics)
Languages : en
Pages : 58

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In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered graph as a subgraph. In particular, we take a step toward confirming a conjecture of Pach and Tardos regarding the number of edges allowed when the forbidden pattern is a tree by establishing an upper bound for a particular ordered graph for which existing techniques have failed. We also generalize a theorem of Geneson by establishing an upper bound on the number of edges allowed if the forbidden graphs fit a generalized notion of a matching.

Author: Paul S. Wenger
Publisher:
ISBN:
Category :
Languages : en
Pages :

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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.

Author: John Adrian Bondy
Publisher: Toronto ; Orlando : Academic Press
ISBN:
Category : Mathematics
Languages : en
Pages : 568

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Author: Marshall Hall
Publisher: Taylor & Francis
ISBN: 9789027705938
Category : Mathematics
Languages : en
Pages : 500

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Proceedings of the NATO Advanced Study Institute, Breukelen, The Netherlands, July 8-30, 1974

Author: Béla Bollobás
Publisher: American Mathematical Soc.
ISBN: 0821807129
Category : Mathematics
Languages : en
Pages : 74

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Book Description
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.

Author: Bela Bollobas
Publisher: Courier Corporation
ISBN: 0486317587
Category : Mathematics
Languages : en
Pages : 512

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Book Description
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.

Author: Frank Harary
Publisher: Courier Dover Publications
ISBN: 0486796841
Category : Mathematics
Languages : en
Pages : 129

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Lectures given in F. Harary's seminar course, University College of London, Dept. of Mathematics, 1962-1963.

Author: Wojciech Samotij
Publisher:
ISBN:
Category :
Languages : en
Pages :

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This dissertation tackles several questions in extremal graph theory and the theory of random graphs. It consists of three more or less independent parts that all fit into one bigger picture -- the meta-problem of describing the structure and properties of large random and pseudo-random graphs. Given a positive constant c, we call an n-vertex graph G c-Ramsey if G does not contain a clique or an independent set of size greater than c*log(n). Since all of the known examples of Ramsey graphs come from various constructions employing randomness, several researchers have conjectured that all Ramsey graphs possess certain pseudo-random properties. We study one such question -- a conjecture of Erdos, Faudree, and Sos regarding the orders and sizes of induced subgraphs of Ramsey graphs. Although we do not fully resolve this conjecture, the main theorem in the first part of this dissertation, joint work with Noga Alon, Jozsef Balogh, and Alexandr Kostochka, significantly improves the previous state-of-the-art result of Alon and Kostochka. For a positive integer n and a real number p in [0,1], one defines the Erdos-Renyi random graph G(n,p) to be the probability distribution on the set of all graphs on the vertex set {1,...,n} such that the probability that a particular pair {i,j} of vertices is an edge in G(n,p) is p, independently of all other pairs. In the second part of this dissertation, we study the behavior of the random graph G(n,p) with respect to the property of containing large trees with bounded maximum degree. Our first main theorem, joint work with Jozsef Balogh, Bela Csaba, and Martin Pei, gives a sufficient condition on p to imply that with probability tending to 1 as n tends to infinity, G(n,p) contains all almost spanning trees with bounded maximum degree, improving a previous result of Alon, Krivelevich, and Sudakov. In the second main theorem of this part, joint work with Jozsef Balogh and Bela Csaba, we show that G(n,p) almost surely contains all almost spanning trees with bounded maximum degree even after an adversary removes asymptotically half of the edges in G(n,p). Given an arbitrary graph H, we say that a graph G is H-free if G does not contain H as a subgraph. Edros, Frankl, and Rodl generalized a famous theorem of Erdos and Stone by proving that for every non-bipartite H, the number of labeled H-free graphs on a fixed n-vertex set, f_n(H), satisfies log_2f_n(H)