Line graph of a hypergraph: Difference between revisions

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TheIn [[graph theory]], particularly in the theory of [[hypergraph]]<nowiki/>s, 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.

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 {{harv|Berge|1989}}.