Combinatorial Nullstellensatz: With Applications to Graph Colouring

Combinatorial Nullstellensatz is a theorem that helps solve combinatorial problems in different areas of mathematics by showing that the coefficient of a certain monomial in a polynomial expansion is nonzero. This book focuses on its applications to graph coloring, particularly three methods for calculating the coefficients: Alon-Tarsi orientation, interpolation formula, and coefficients as the permanents of matrices. It is a valuable reference book for graduate courses in mathematics.

Format: Hardback
Length: 134 pages
Publication date: 01 June 2021
Publisher: Taylor & Francis Ltd

Combinatorial Nullstellensatz is a groundbreaking theorem in algebra developed by Noga Alon to address complex combinatorial problems across various branches of mathematics. This comprehensive book delves into the practical applications of this theorem in the realm of graph coloring. A crucial aspect of applying Combinatorial Nullstellensatz is demonstrating that the coefficient of a specific monomial in the expansion of a polynomial is nonzero. The central focus of the book revolves around three methods for calculating these coefficients:

Alon-Tarsi Orientation: This method involves proving that a graph possesses a specific orientation with a given maximum out-degree, while the number of even Eulerian sub-digraphs differs from the number of odd Eulerian sub-digraphs. Specifically, it is used to show that a graph with an edge set that decomposes into a Hamilton cycle and vertex-disjoint triangles is 3-choosable. Moreover, it demonstrates that every planar graph possesses a matching that, upon deletion, results in a 4-choosable graph.

Interpolation Formula for the Coefficient: This method is particularly employed to demonstrate 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 finds application in the study of the list version of vertex-edge weighting and serves to show that every graph is (2,3)-choosable. It is particularly well-suited as a reference book for advanced graduate courses in mathematics.

Weight: 306g
Dimension: 145 x 223 x 17 (mm)
ISBN-13: 9780367686949