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Combinatorial Nullstellensatz

With Applications to Graph Colouring

By: R. Balakrishnan, Xuding Zhu

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Hardcover | 1 June 2021 | Edition Number 1

At a Glance

Hardcover
152 Pages

Dimensions(cm)
14.5 x 22.3 x 1.7

Hardcover

$214.75

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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.

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