Line graphs of hypergraphs

The line graph of a hypergraph is the 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 (Berge 1989).

Beineke (1968) characterized line graphs of graphs by a list of 9 forbidden induced subgraphs. (See the article on line graphs.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any k ≥ 3, and Lovász (1977) showed there is no such characterization by a finite list if k = 3.

Krausz (1943) characterized line graphs of graphs in terms of clique 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 Berge (1989).

A global characterization of Krausz type for the line graphs of k-uniform linear hypergraphs for any k ≥ 3 was given by Naik et al. (1980). 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. Metelsky & Tyshkevich (1997) and Jacobson, Kézdy & Lehel (1997) improved this bound to 19. At last Skums, Suzdal' & Tyshkevich (2005) reduced this bound to 16. Metelsky & Tyshkevich (1997) 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.

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