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 t
