graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, particularly in the theory of

hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) wh ...

s, the line graph of a hypergraph , denoted , is the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

whose

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

set is the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of the hyperedges of , with two vertices adjacent in when their corresponding hyperedges have a nonempty

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

in . In other words, is the

intersection graph In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types o ...

of a family of

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...

s. It is a

generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...

of the

line graph In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every ...

of a

. Questions about line graphs of hypergraphs are often generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size is called . (A 2-uniform hypergraph is a graph). In hypergraph theory, it is often natural to require that hypergraphs be . Every graph is the line graph of some hypergraph, but, given a fixed edge size , not every graph is a line graph of some hypergraph. A main problem is to characterize those that are, for each . A hypergraph is linear if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some hypergraph, but of some linear hypergraph .

Line graphs of ''k''-uniform hypergraphs, ''k'' ≥ 3

characterized line graphs of graphs by a list of 9

forbidden induced subgraph In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidde ...

s. (See the article on

s.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any ''k'' ≥ 3, and showed there is no such characterization by a finite list if ''k'' = 3. characterized line graphs of graphs in terms of

clique A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popular ...

covers. (See Line Graphs.) A global characterization of Krausz type for the line graphs of ''k''-uniform hypergraphs for any ''k'' ≥ 3 was given by .

Line graphs of ''k''-uniform linear hypergraphs, ''k'' ≥ 3

A global characterization of Krausz type for the line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3 was given by . At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. and improved this bound to 19. At last reduced this bound to 16. also proved that, if ''k'' > 3, no such finite list exists for linear ''k''-uniform hypergraphs, no matter what lower bound is placed on the degree. The difficulty in finding a characterization of linear ''k''-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for ''m'' > 0, consider a chain of ''m''

diamond graph In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph minus one edge. The diamond graph has radius 1, diameter 2, girth 3, chromat ...

s such that the consecutive diamonds share vertices of degree two. For ''k'' ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs of as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke's of line graphs of graphs. There are some interesting characterizations available for line graphs of linear ''k''-uniform hypergraphs due to various authors (, , , and ) under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least ''k''³-2''k''²+1 in is reduced to 2''k''²-3''k''+1 in and to characterize line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3. The complexity of recognizing line graphs of linear ''k''-uniform hypergraphs without any constraint on minimum degree (or minimum edge-degree) is not known. For ''k'' = 3 and minimum degree at least 19, recognition is possible in polynomial time ( and ). reduced the minimum degree to 10. There are many interesting open problems and conjectures in Naik et al., Jacoboson et al., Metelsky et al. and Zverovich.

Disjointness graph

The disjointness graph of a hypergraph ''H'', denoted D(''H''), is the graph whose vertex set is the set of the hyperedges of ''H'', with two vertices adjacent in D(''H'') when their corresponding hyperedges are ''disjoint'' in ''H''. In other words, D(''H'') is the

complement graph In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a ...

of L(''H''). A

in D(''H'') corresponds to an independent set in L(''H''), and vice versa.

References

*. *. Translated from the French. *. *. *. (In Hungarian, with French abstract.) *. *. *. *. *. *. *. *{{citation , first = Vitaly I. , last = Voloshin , title = Introduction to Graph and Hypergraph Theory , location = New York , publisher =

Nova Science Publishers, Inc. Nova Science Publishers is an academic publisher of books, encyclopedias, handbooks, e-books and journals, based in Hauppauge, New York. It was founded in 1985. A prolific publisher of books, Nova has received criticism from librarians for not ...

, year = 2009 , mr=2514872 Graph families Hypergraphs