Chromaticity Of Hypergraphs

The coloring the vertices of a graph is one of the fundamental concepts of graph theory. It is widely believed that coloring was first mentioned in 1852 when Francis Guthrie asked if four colors are enough to color any geographic map in such a way that no two countries sharing a common border would have the same color. If we denote the countries by points in the plane and connect each pair of points that correspond to two countries with a common border by a curve, we obtain a planar graph. The celebrated four color problem asks if every planer graph can be colored with 4 colors. The four color problem became one of the most famous problem in discrete mathematics of the 20th century. This has spawned the development of many useful tools for solving graph coloring problems. The coloring of hypergraphs started in 1966 when P. Erdos and A. Hajnal introduced the notion of coloring of a hypergraph and obtained the first important results. Since then many results in graph colorings have been extended to hyper- graphs. This work focuses on the chromatic polynomial and chromatic uniqueness of hypergraphs.

The theory of graph coloring has existed for more than 150 years. Historically, graph coloring involved finding the minimum number of colors to be assigned to the vertices so that adjacent vertices would have different colors. From this modest beginning, the theory has become central in discrete mathematics with many contemporary generalizations and applications. Generalization of graph coloring-type problems to mixed hypergraphs brings many new dimensions to the theory ofcolorings. A main feature of this book is that in the case of hypergraphs, there exist problems on both the minimum and the maximum number of colors. This feature pervades the theory, methods, algorithms, and applications of mixed hypergraph coloring. The book has broad appeal. It will be of interest to bothpure and applied mathematicians, particularly those in the areas of discrete mathematics, combinatorial optimization, operations research, computer science, software engineering, molecular biology, and related businesses and industries. It also makes a nice supplementary text for courses in graph theory and discrete mathematics. This is especially useful for students in combinatorics and optimization. Since the area is new, students will have the chance at this stage to obtain results that maybecome classic in the future.

Numerous well-presented and important papers from the conference are gathered in the proceedings for the purpose of pointing directions for useful future research in diverse areas of mathematics including algebraic geometry, analysis, commutative algebra, complex analysis, discrete mathematics, dynamical systems, number theory and topology. Several papers on computational and applied mathematics such as wavelet analysis, quantum mechanics, piecewise linear modeling, cosmological models of super symmetry, fluid dynamics, interpolation theory, optimization, ergodic theory and games theory are also presented.

A generalization of the circular chromatic number to hypergraphs is devel-oped. Circular colourings of graphs and hypergraphs are first discussed and it is shown that the circular chromatic number of a graph is the same regard-less of whether the hypergraph or graph definition is used. After presenting a few basic results, some examples of circular chromatic numbers of various families of hypergraphs are given. Subsequently, the concepts of the star chromatic number and the arc chromatic number are introduced. Specif?ically, both numbers are shown to be equivalent to the circular chromatic number. Finally the relationship between the imbalance of a hypergraph and the circular chromatic number is explored and a classical result of Minty is deduced.

A generalization of the circular chromatic number to hypergraphs is devel-oped. Circular colourings of graphs and hypergraphs are first discussed and it is shown that the circular chromatic number of a graph is the same regard-less of whether the hypergraph or graph definition is used. After presenting a few basic results, some examples of circular chromatic numbers of various families of hypergraphs are given. Subsequently, the concepts of the star chromatic number and the arc chromatic number are introduced. Specif?ically, both numbers are shown to be equivalent to the circular chromatic number. Finally the relationship between the imbalance of a hypergraph and the circular chromatic number is explored and a classical result of Minty is deduced.

A generalization of the circular chromatic number to hypergraphs is devel-oped. Circular colourings of graphs and hypergraphs are first discussed and it is shown that the circular chromatic number of a graph is the same regard-less of whether the hypergraph or graph definition is used. After presenting a few basic results, some examples of circular chromatic numbers of various families of hypergraphs are given. Subsequently, the concepts of the star chromatic number and the arc chromatic number are introduced. Specifically, both numbers are shown to be equivalent to the circular chromatic number. Finally the relationship between the imbalance of a hypergraph and the circular chromatic number is explored and a classical result of Minty is deduced.

"This is the first book to comprehensively cover chromatic polynomials of graphs. It includes most of the known results and unsolved problems in the area of chromatic polynomials. Dividing the book into three main parts, the authors take readers from the rudiments of chromatic polynomials to more complex topics: the chromatic equivalence classes of graphs and the zeros and inequalities of chromatic polynomials. The early material is well suited to a graduate level course while the latter parts will be an invaluable resource for postgraduate students and researchers in combinatorics and graph theory."--BOOK JACKET.

Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory?some of which are nontrivial?have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors. Tensor Analysis: Spectral Theory and Special Tensors is unique in that it is the first book on these three subject areas: spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors.