We can define a weaker notion of hypergraph acyclicity, later termed α-acyclicity. Hypergraphs can be viewed as. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. In some literature edges are referred to as hyperlinks or connectors. Fields Institute Monographs, American Mathematical Society, 2002. Next, it was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. An Application to an Extremal Problem.
In theory, hypergraphs are called simple games voting games ; this notion is applied to solve problems in. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k- uniform hypergraph is a hypergraph such that all its hyperedges have size k. In particular, there is no transitive closure of set membership for such hypergraphs. While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. Some mixed hypergraphs are uncolorable for any number of colors. Hypergraphs for which there exists a coloring using up to k colors are referred to as k-colorable.
It is with this motivation that we have tried in this book to present what has seemed to us to be the most significant work on hypergraphs. We show that hypergraphs can be extended to face-hypergraphs in a natural way and use tools from topological graph theory, the theory of hypergraphs, and design theory to obtain general bounds for the coloring and choosability problems. A hypergraph is k-colorable k-choosable if there is a coloring of its vertices from a set of k colors from every assignment of lists of size k to its vertices such that no edge is monochromatic. Similarly, a hypergraph is edge-transitive if all edges are symmetric. . Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. A hypergraph is a graph in which generalized edges called hyperedges may connect more than two Berge, C. Berge was very much a product of the old school and 'clean' and 'pretty' proofs mattered to him almost as much as the proof itself. This book presents what seems to be the most significant work on hypergraphs. When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality.
A general criterion for uncolorability is unknown. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on ad infinitum. The face-hypergraph, H G , of a graph G embedded in a surface has vertex set V G , and every face of G corresponds to an edge of H G consisting of the vertices incident to the face. In Hungarian, with French abstract. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs.
The partial hypergraph is a hypergraph with some edges removed. In essence, every edge is just an internal node of a tree or , and vertices are the leaf nodes. Efficient and Scalable are also important for processing large scale hypergraphs in machine learning tasks. H is k-regular if every vertex has degree k. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.
For the past forty years, Graph Theory has proved to be an extremely useful tool for solving combinatorial problems, in areas as diverse as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. In hypergraph theory, it is often natural to require that hypergraphs be k-uniform. There are two variations of this generalization. Besides, α-acyclicity is also related to the expressiveness of the of. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs.
Conversely, every collection of trees can be understood as this generalized hypergraph. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. Computing the transversal hypergraph has applications in , in , and in several fields of such as , , the , , and computer. The book is an excellent source for both student and researcher. Thus a proper coloring of a face-hypergraph corresponds to a vertex coloring of the underlying graph such that no face is monochromatic. The 2-section or clique graph, representing graph, primal graph, Gaifman graph of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.
Harary, Addison Wesley, 1969 p. In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes. Hypergraphs have many other names. The idea of looking at a family of sets from this standpoint took shape around 1960. A hypergraph is linear if each pair of hyperedges intersects in at most one vertex.