1st Edition

# Combinatorial Nullstellensatz With Applications to Graph Colouring

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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.

**Some definitions and notations. Combinatorial Nullstellensatz. **Introduction. An application of CNS to additive number theory. Application of CNS to geometry. CNS and a subgraph problem. 0-1 vectors in a hyperplane.** Alon-Tarsi Theorem and its Applications. **Alon-Tarsi Theorem. Bipartite graphs and acyclic orientations. The Cartesian product of a path and an odd cycle. A solution to a problem of Erd˝os. Bound for *AT *(*G*) in terms of degree. Planar graphs. Planar graph minus a matching. Discharging method. Hypergraph colouring. Paintability of graphs. **Generalizations of CNS and applications. **Number of nonzero points. Multisets. Coefficient of a highest degree monomial. Calculation of *N*** _{S}**(

**a**). Alon-Tarsi number of

*K*

_{2*n}

*and cycle powers. Alon-Tarsi numbers of toroidal grids. List colouring of line graphs.*

*r*-regular planar graphs. Complete graphs

*K*+1 for odd prime

_{p}*p.*Jaeger’s Conjecture.

**Permanent and vertex-edge weighting.**Permanent as the coefficient. Edge weighting and total weighting. Polynomial associated to total weighting. Permanent index. Trees with an even number of edges. Complete graphs. Every graph is (2

*,*3)-choosable.

### Biography

**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.