Line graph of a hypergraph: Difference between revisions

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In [[graph theory]], particularly in the theory of [[hypergraph]]<nowiki/>s, the '''line graph of a hypergraph''' ''H'', denoted L(''H''), is the [[graph (discrete mathematics)|graph]] whose vertex set is the set of the hyperedges of the hypergraph''H'', with two hyperedgesvertices adjacent in L(''H'') when theytheir corresponding hyperedges have a nonempty intersection in ''H''. In other words, the line graph of a hypergraphL(''H'') is the [[intersection graph]] of a family of finite sets. It is a generalization of the [[line graph]] of a graph.

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 ''k'' is called ''k'''''-uniform'''. (A 2-uniform hypergraph is a graph). In hypergraph theory, it is often natural to require that hypergraphs be ''k''-uniform. Every graph is the line graph of some hypergraph, but, given a fixed edge size ''k'', not every graph is a line graph of some ''k''-uniform hypergraph. A main problem is to characterize those that are, for each ''k'' ≥ 3.

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 .<ref>{{harvCite book|last=Berge, C. (Claude)|first=|url=http://worldcat.org/oclc/898849714|title=Hypergraphs : combinatorics of finite sets|date=1989|publisher=North Holland|others=MR 1013569|year=|isbn=0-444-87489-5|___location=|pages=|language=Translated from the French.|oclc=898849714}}.</ref>

==Line graphs of ''k''-uniform hypergraphs, ''k'' &ge; 3==

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There are many interesting open problems and conjectures in Naik et al., Jacoboson et al., Metelsky et al. and Zverovich.

== Disjointness graph ==

==References==

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''.<ref>{{Cite journal|last=Meshulam|first=Roy|date=2001-01-01|title=The Clique Complex and Hypergraph Matching|url=https://doi.org/10.1007/s004930170006|journal=Combinatorica|language=en|volume=21|issue=1|pages=89–94|doi=10.1007/s004930170006|issn=1439-6912}}</ref> In other words, D(''H'') is the [[complement graph]] of L(''H''). An [[Clique (graph theory)|clique]] in D(''H'') corresponds to an independent set in L(''H''), and vice-versa.

== References ==

*{{citation

| first = L. W. | last = Beineke

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| editor3-first = H. | editor3-last = Walther

| publisher = Teubner | ___location = Leipzig | pages = 17–23 | year = 1968}}.

*{{citation

| first = C. | last = Berge | authorlink = Claude Berge

| title = Hypergraphs: Combinatorics of Finite Sets

| ___location = Amsterdam | publisher = North-Holland | year = 1989

|mr=1013569}}. Translated from the French.

*{{citation