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Combinatorial Nullstellensatz : With Applications to Graph Colouring.
・ISBN 978-0-367-68694-9 hard GB£ 67.99
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電子版あり 大学・学術機関向け電子ブック（eBook）ISBN 978-1-003-13867-9
| 著者・編者 | Zhu, Xuding / Balakrishnan, R., |
|---|---|
| 出版社 | (Chapman & Hall/CRC, UK) |
| 出版年月 | 2021 |
| ページ数 | 134 pp. |
| 言語 | ENG |
| ニュース番号 | ＜M25-4065＞ |
解説
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.