Combinatorial Nullstellensatz

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Combinatorial Nullstellensatz

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Author : Xuding Zhu
language : en
Publisher: CRC Press
Release Date : 2021-06-01

Combinatorial Nullstellensatz written by Xuding Zhu and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-01 with Mathematics categories.

Combinatorial Nullstellensatz is a novel theorem in algebra introduced by Noga Alon to tackle combinatorial problems in diverse areas of mathematics. This book focuses on the applications of this theorem to graph colouring. A key step in the applications of Combinatorial Nullstellensatz is to show that the coefficient of a certain monomial in the expansion of a polynomial is nonzero. The major part of the book concentrates on three methods for calculating the coefficients: Alon-Tarsi orientation: The task is to show that a graph has an orientation with given maximum out-degree and for which the number of even Eulerian sub-digraphs is different from the number of odd Eulerian sub-digraphs. In particular, this method is used to show that a graph whose edge set decomposes into a Hamilton cycle and vertex-disjoint triangles is 3-choosable, and that every planar graph has a matching whose deletion results in a 4-choosable graph. Interpolation formula for the coefficient: This method is in particular used to show that toroidal grids of even order are 3-choosable, r-edge colourable r-regular planar graphs are r-edge choosable, and complete graphs of order p+1, where p is a prime, are p-edge choosable. Coefficients as the permanents of matrices: This method is in particular used in the study of the list version of vertex-edge weighting and to show that every graph is (2,3)-choosable. It is suited as a reference book for a graduate course in mathematics.

Combinatorial Nullstellensatz

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Author : Xuding Zhu
language : en
Publisher: CRC Press
Release Date : 2021-05-31

Combinatorial Nullstellensatz written by Xuding Zhu and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-31 with Mathematics categories.

Combinatorial Nullstellensatz is a novel theorem in algebra introduced by Noga Alon to tackle combinatorial problems in diverse areas of mathematics. This book focuses on the applications of this theorem to graph colouring. A key step in the applications of Combinatorial Nullstellensatz is to show that the coefficient of a certain monomial in the expansion of a polynomial is nonzero. The major part of the book concentrates on three methods for calculating the coefficients: Alon-Tarsi orientation: The task is to show that a graph has an orientation with given maximum out-degree and for which the number of even Eulerian sub-digraphs is different from the number of odd Eulerian sub-digraphs. In particular, this method is used to show that a graph whose edge set decomposes into a Hamilton cycle and vertex-disjoint triangles is 3-choosable, and that every planar graph has a matching whose deletion results in a 4-choosable graph. Interpolation formula for the coefficient: This method is in particular used to show that toroidal grids of even order are 3-choosable, r-edge colourable r-regular planar graphs are r-edge choosable, and complete graphs of order p+1, where p is a prime, are p-edge choosable. Coefficients as the permanents of matrices: This method is in particular used in the study of the list version of vertex-edge weighting and to show that every graph is (2,3)-choosable. It is suited as a reference book for a graduate course in mathematics.

Applications Of The Combinatorial Nullstellensatz On Bipartite Graphs

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Author : Timothy M. Brauch
language : en
Publisher:
Release Date : 2009

Applications Of The Combinatorial Nullstellensatz On Bipartite Graphs written by Timothy M. Brauch and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Combinatorial analysis categories.

The Combinatorial Nullstellensatz can be used to solve certain problems in combinatorics. However, one of the major complications in using the Combinatorial Nullstellensatz is ensuring that there exists a nonzero monomial. This dissertation looks at applying the Combinatorial Nullstellensatz to finding perfect matchings in bipartite graphs. The first two chapters provide background material covering topics such as linear algebra, group theory, graph theory and even the discrete Fourier transform. New results start in the third chapter, showing that the Combinatorial Nullstellensatz can be used to solve the problem of finding perfect matchings in bipartite graphs. Using the Combinatorial Nullstellensatz also allows for a vice use of matroid intersection to find the nonzero monomial. By also applying the uncertainty principle, the number of perfect matchings in a bipartite graph can be bound. The fourth chapter examines properties of the polynomials created in the use of the Combinatorial Nullstellensatz to find perfect matchings in bipartite graphs. Many of the properties of the polynomials have analogous properties for the transforms of the polynomials, which are also examined. These properties often relate back to the structure of the graph which gave rise to the polynomial. The fifth chapter provides an application of the results. Since finding a nonzero monomial can be difficult and the polynomials created in this dissertation give polynomials with such a nonzero monomial the application shows how certain polynomials can be rewritten in terms of the matching polynomials. Such a rewriting may permit an easy method to find a nonzero monomial so that the Combinatorial Nullstellensatz can be applied to the polynomial. Finally, the fifth chapter concludes with some open problems that may be areas of further research.

Applications Of The Combinatorial Nullstellensatz

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Author : Nagy Zoltán Lóránt
language : en
Publisher:
Release Date : 2014

Applications Of The Combinatorial Nullstellensatz written by Nagy Zoltán Lóránt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.

Computational Aspects Of The Combinatorial Nullstellensatz Method Via A Polynomial Approach To Matrix And Hypermatrix Algebra

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Author : Edinah K. Gnang
language : en
Publisher:
Release Date : 2013

Computational Aspects Of The Combinatorial Nullstellensatz Method Via A Polynomial Approach To Matrix And Hypermatrix Algebra written by Edinah K. Gnang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Combinatorial analysis categories.

Combinatorial Algorithms

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Author : Ljiljana Brankovic
language : en
Publisher: Springer
Release Date : 2018-04-19

Combinatorial Algorithms written by Ljiljana Brankovic and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-04-19 with Computers categories.

This book constitutes the refereed post-conference proceedings of the 28th International Workshopon Combinatorial Algorithms, IWOCA 2017, held in Newcastle, NSW, Australia, in July 2017.The 30 regular papers presented in this volume together with 5 invited talks were carefully reviewed and selected from 55 submissions. They were organized in topical sessions named: approximation algorithms and hardness; computational complexity; computational geometry; graphs and combinatorics; graph colourings, labellings and power domination; heuristics; mixed integer programming; polynomial algorithms; privacy; and string algorithms.

Extremal Combinatorics

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Author : Stasys Jukna
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-08-31

Extremal Combinatorics written by Stasys Jukna and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-08-31 with Computers categories.

This book is a concise, self-contained, up-to-date introduction to extremal combinatorics for nonspecialists. There is a strong emphasis on theorems with particularly elegant and informative proofs, they may be called gems of the theory. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science: methods of extremal set theory, the linear algebra method, the probabilistic method, and fragments of Ramsey theory. No special knowledge in combinatorics or computer science is assumed – the text is self-contained and the proofs can be enjoyed by undergraduate students in mathematics and computer science. Over 300 exercises of varying difficulty, and hints to their solution, complete the text. This second edition has been extended with substantial new material, and has been revised and updated throughout. It offers three new chapters on expander graphs and eigenvalues, the polynomial method and error-correcting codes. Most of the remaining chapters also include new material, such as the Kruskal—Katona theorem on shadows, the Lovász—Stein theorem on coverings, large cliques in dense graphs without induced 4-cycles, a new lower bounds argument for monotone formulas, Dvir's solution of the finite field Kakeya conjecture, Moser's algorithmic version of the Lovász Local Lemma, Schöning's algorithm for 3-SAT, the Szemerédi—Trotter theorem on the number of point-line incidences, surprising applications of expander graphs in extremal number theory, and some other new results.

Issues In Logic Probability Combinatorics And Chaos Theory 2011 Edition

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Author :
language : en
Publisher: ScholarlyEditions
Release Date : 2012-01-09

Issues In Logic Probability Combinatorics And Chaos Theory 2011 Edition written by and has been published by ScholarlyEditions this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-09 with Mathematics categories.

Issues in Logic, Probability, Combinatorics, and Chaos Theory: 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about Logic, Probability, Combinatorics, and Chaos Theory. The editors have built Issues in Logic, Probability, Combinatorics, and Chaos Theory: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Logic, Probability, Combinatorics, and Chaos Theory in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Logic, Probability, Combinatorics, and Chaos Theory: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Applied Hilbert S Nullstellensatz For Combinatorial Problems

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Author : Julian Romero Barbosa
language : en
Publisher:
Release Date : 2016

Applied Hilbert S Nullstellensatz For Combinatorial Problems written by Julian Romero Barbosa and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Combinatorial optimization categories.

Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \textit{stable set} of a given size, finding the \textit{chromatic number} of a graph or more generally, determining the feasibility of an \textit{Integer Programming problem} are classical examples of this. In this thesis we study a powerful tool from Algebraic Geometry, called \textit{Hilbert's Nullstellensatz}. It characterizes the \textit{infeasibility} of a system of polynomial equations by the \textit{feasibility} of a possibly very large system of \textit{linear equations}. The solutions to this linear system provide \textit{certificates} for the infeasibility of the polynomial system, called \textit{Nullstellensatz Certificates}. In this thesis we focus on the study of Nullstellensatz Certificates for the existence of \textit{proper colorings} of graphs. We use basic ideas from \textit{duality theory} to determine various properties of the Nullstellensatz Certificates. We give new proofs to several known results in the current literature and present some new results that shed some light on the relationship between the sparsity of a graph and the \textit{size} of the Nullstellensatz Certificates for \textit{$k$-colorability}.

Combinatorial Mathematics

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Author : Douglas B. West
language : en
Publisher: Cambridge University Press
Release Date : 2020-07-16

Combinatorial Mathematics written by Douglas B. West and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-16 with Mathematics categories.

This is the most readable and thorough graduate textbook and reference for combinatorics, covering enumeration, graphs, sets, and methods.