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 ''H'', with two vertices adjacent in L(''H'') when their corresponding hyperedges have a nonempty intersection in ''H''. In other words, L(''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.