On Binary And Regular Matroids Without Small Minors
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Author | : Dillon Mayhew |
Publisher | : American Mathematical Soc. |
Total Pages | : 110 |
Release | : 2010 |
Genre | : Mathematics |
ISBN | : 0821848267 |
The authors give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.
Author | : Kayla Davis Harville |
Publisher | : |
Total Pages | : 222 |
Release | : 2013 |
Genre | : |
ISBN | : |
The results of this dissertation consist of excluded-minor results for Binary Matroids and excluded-minor results for Regular Matroids. Structural theorems on the relationship between minors and k- sums of matroids are developed here in order to provide some of these characterizations. Chapter 2 of the dissertation contains excluded-minor results for Binary Matroids. The first main result of this dissertation is a characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes results of Mayhew and Royle [18] for binary matroids and results of Dirac [8] and Lovasz [15] for graphs. The results of this chapter are then extended from the class of internally 4-connected matroids to the class of 3-connected matroids. Chapter 3 of the dissertation contains the second main result, a decomposition theorem for regular matroids without certain minors. This decomposition theorem is used to obtain excluded-minor results for Regular Matroids. Wagner, Lovasz, Oxley, Ding, Liu, and others have characterized many classes of graphs that are H- free for graphs H with at most twelve edges (see [7]). We extend several of these excluded-minor characterizations to regular matroids in Chapter 3. We also provide characterizations of regular matroids excluding several graphic matroids such as the octahedron, cube, and the Mobius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation.
Author | : Hirokazu Nishimura |
Publisher | : Springer Science & Business Media |
Total Pages | : 238 |
Release | : 2009-04-21 |
Genre | : Mathematics |
ISBN | : 3764385731 |
Matroid theory was invented in the middle of the 1930s by two mathematicians independently, namely, Hassler Whitney in the USA and Takeo Nakasawa in Japan. Whitney became famous, but Nakasawa remained anonymous until two decades ago. He left only four papers to the mathematical community, all of them written in the middle of the 1930s. It was a bad time to have lived in a country that had become as eccentric as possible. Just as Nazism became more and more flamboyant in Europe in the 1930s, Japan became more and more esoteric and fanatical in the same time period. This book explains the little that is known about Nakasawa’s personal life in a Japan that had, among other failures, lost control over its military. This book contains his four papers in German and their English translations as well as some extended commentary on the history of Japan during those years. The book also contains 14 photos of him or his family. Although the veil of mystery surrounding Nakasawa’s life has only been partially lifted, the work presented in this book speaks eloquently of a tragic loss to the mathematical community.
Author | : KUNG |
Publisher | : Springer Science & Business Media |
Total Pages | : 400 |
Release | : 2013-11-09 |
Genre | : Mathematics |
ISBN | : 1468491997 |
by Gian-Carlo Rota The subjects of mathematics, like the subjects of mankind, have finite lifespans, which the historian will record as he freezes history at one instant of time. There are the old subjects, loaded with distinctions and honors. As their problems are solved away and the applications reaped by engineers and other moneymen, ponderous treatises gather dust in library basements, awaiting the day when a generation as yet unborn will rediscover the lost paradise in awe. Then there are the middle-aged subjects. You can tell which they are by roaming the halls of Ivy League universities or the Institute for Advanced Studies. Their high priests haughtily refuse fabulous offers from eager provin cial universities while receiving special permission from the President of France to lecture in English at the College de France. Little do they know that the load of technicalities is already critical, about to crack and submerge their theorems in the dust of oblivion that once enveloped the dinosaurs. Finally, there are the young subjects-combinatorics, for instance. Wild eyed individuals gingerly pick from a mountain of intractable problems, chil dishly babbling the first words of what will soon be a new language. Child hood will end with the first Seminaire Bourbaki. It could be impossible to find a more fitting example than matroid theory of a subject now in its infancy. The telltale signs, for an unfailing diagnosis, are the abundance of deep theorems, going together with a paucity of theories.
Author | : Gary Gordon |
Publisher | : Cambridge University Press |
Total Pages | : 406 |
Release | : 2012-08-02 |
Genre | : Mathematics |
ISBN | : 1139536087 |
Matroid theory is a vibrant area of research that provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to any mathematician interested in the geometric approach to matroids. Written in a friendly, fun-to-read style and developed from the authors' own undergraduate courses, the book is ideal for students. Beginning with a basic introduction to matroids, the book quickly familiarizes the reader with the breadth of the subject, and specific examples are used to illustrate the theory and to help students see matroids as more than just generalizations of graphs. Over 300 exercises are included, with many hints and solutions so students can test their understanding of the materials covered. The authors have also included several projects and open-ended research problems for independent study.
Author | : Gary Gordon |
Publisher | : Cambridge University Press |
Total Pages | : 411 |
Release | : 2012-08-02 |
Genre | : Language Arts & Disciplines |
ISBN | : 0521145686 |
This friendly introduction helps undergraduate students understand and appreciate matroid theory and its connections to geometry.
Author | : Henry H. Crapo |
Publisher | : MIT Press (MA) |
Total Pages | : 350 |
Release | : 1970 |
Genre | : Mathematics |
ISBN | : |
A major aim of this book is to present the theory of combinatorial geometry in a form accessible to mathematicians working in disparate subjects.
Author | : Simon R. Blackburn |
Publisher | : Cambridge University Press |
Total Pages | : 387 |
Release | : 2013 |
Genre | : Combinatorial analysis |
ISBN | : 1107651956 |
Surveys of recent important developments in combinatorics covering a wide range of areas in the field.
Author | : Bozzano G Luisa |
Publisher | : Elsevier |
Total Pages | : 1121 |
Release | : 1995-12-11 |
Genre | : Computers |
ISBN | : 0080933351 |
Handbook of Combinatorics, Volume 1 focuses on basic methods, paradigms, results, issues, and trends across the broad spectrum of combinatorics. The selection first elaborates on the basic graph theory, connectivity and network flows, and matchings and extensions. Discussions focus on stable sets and claw free graphs, nonbipartite matching, multicommodity flows and disjoint paths, minimum cost circulations and flows, special proof techniques for paths and circuits, and Hamilton paths and circuits in digraphs. The manuscript then examines coloring, stable sets, and perfect graphs and embeddings and minors. The book takes a look at random graphs, hypergraphs, partially ordered sets, and matroids. Topics include geometric lattices, structural properties, linear extensions and correlation, dimension and posets of bounded degree, hypergraphs and set systems, stability, transversals, and matchings, and phase transition. The manuscript also reviews the combinatorial number theory, point lattices, convex polytopes and related complexes, and extremal problems in combinatorial geometry. The selection is a valuable reference for researchers interested in combinatorics.
Author | : Alan Frieze |
Publisher | : Cambridge University Press |
Total Pages | : 483 |
Release | : 2016 |
Genre | : Mathematics |
ISBN | : 1107118506 |
The text covers random graphs from the basic to the advanced, including numerous exercises and recommendations for further reading.