Combinatorial Nullstellensatz: With Applications to Graph Colouring
暫譯: 組合空集定理：及其在圖形著色中的應用

Zhu, Xuding, Balakrishnan, R.

Combinatorial Nullstellensatz: With Applications to Graph Colouring
  • 出版商: CRC
  • 出版日期: 2021-06-01
  • 售價: $2,740
  • 貴賓價: 9.5 折 $2,603
  • 語言: 英文
  • 頁數: 134
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 0367686945
  • ISBN-13: 9780367686949

    • 海外代購書籍(需單獨結帳)

商品描述

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:

  1. 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.

  2. 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.

  3. 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.

商品描述(中文翻譯)

組合零點定理（Combinatorial Nullstellensatz）是由 Noga Alon 提出的新穎代數定理，旨在解決數學各個領域中的組合問題。本書專注於該定理在圖著色中的應用。組合零點定理應用的一個關鍵步驟是顯示多項式展開中某個單項式的係數非零。本書的主要部分集中於計算係數的三種方法：

1. Alon-Tarsi 定向：任務是顯示一個圖具有給定的最大出度的定向，並且其偶數歐拉子有向圖的數量與奇數歐拉子有向圖的數量不同。特別地，這種方法用於顯示一個邊集分解為哈密頓迴圈和頂點不相交三角形的圖是 3-可選的，並且每個平面圖都有一個匹配，其刪除結果是一個 4-可選的圖。

2. 係數的插值公式：這種方法特別用於顯示偶數階的環狀網格是 3-可選的，r-邊著色的 r-正則平面圖是 r-邊可選的，以及階數為 p+1 的完全圖（其中 p 是質數）是 p-邊可選的。

3. 係數作為矩陣的行列式：這種方法特別用於研究頂點-邊加權的列表版本，並顯示每個圖都是 (2,3)-可選的。

本書適合作為數學研究生課程的參考書。

作者簡介

Xuding Zhu is currently a Professor of Mathematics, director of the Center for Discrete Mathematics at Zhejiang Normal University, China. His fields of interests are: Combinatorics and Graph Colouring. He published more than 260 research papers and served on the editorial board of SIAM Journal on Discrete Mathematics, Journal of Graph Theory, European Journal of Combinatorics, Electronic Journal of Combinatorics, Discrete Mathematics, Contribution to Discrete Mathematics, Discussion. Math. Graph Theory, Bulletin of Academia Sinica and Taiwanese Journal of Mathematics.​

R. Balakrishnan is currently an Adjunct Professor of Mathematics at Bharathidasan University, Triuchirappalli, India. His fields of interests are: Algebraic Combinatorics and Graph Colouring. He is an author of three other books, one in Graph Theory and the other two in Discrete Mathematics. He is also one of the founders of the Ramanujan Mathematical Society and the Academy of Discrete Mathematics and Applications and currently an Editor-in-Chief of the Indian Journal of Discrete Mathematics.​

作者簡介(中文翻譯)

朱旭丁目前是中國浙江師範大學的數學教授，並擔任離散數學中心的主任。他的研究領域包括：組合數學和圖著色。他發表了超過260篇研究論文，並擔任《SIAM 雜誌：離散數學》、《圖論雜誌》、《歐洲組合數學雜誌》、《電子組合數學雜誌》、《離散數學》、《離散數學貢獻》、《討論數學圖論》、《中央研究院公報》和《台灣數學雜誌》的編輯委員會成員。​

R. Balakrishnan目前是印度特里奇拉帕利的巴哈拉迪丹大學的數學兼任教授。他的研究領域包括：代數組合數學和圖著色。他是另外三本書的作者，其中一本是關於圖論，另外兩本是關於離散數學。他也是拉馬努金數學學會和離散數學及應用學會的創始人之一，目前擔任《印度離散數學雜誌》的主編。​

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