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The '''line graph of a hypergraph''' is the [[graph (discrete mathematics)|graph]] whose vertex set is the set of the hyperedges of the [[hypergraph]], with two hyperedges adjacent when they have a nonempty intersection. In other words, the line graph of a hypergraph 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.
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| first3 = Jeno | last3 = Lehel
| title = Recognizing intersection graphs of linear uniform hypergraphs
| journal = [[Graphs and Combinatorics]] | volume = 13 | pages = 359–367 | year = 1997
|mr=1485929 | doi = 10.1007/BF03353014}}.
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| first = Igor E. | last = Zverovich
| title = A solution to a problem of Jacobson, Kézdy and Lehel
| journal = [[Graphs and Combinatorics]] | volume = 20 | issue = 4 | year = 2004 | pages = 571–577
|mr=2108401 | doi = 10.1007/s00373-004-0572-1}}.
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