Line graph of a hypergraph

The line graph of a hypergraph is the graph whose vertex set is the set of the hyperedges of the hypergraph, with two edges adjacent when they have 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.

Line graphs of "k"-uniform hypergraphs, "k" ≥ 3

harvtxt|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 harvtxt|Lóvász|1977 showed there is no such characterization by a finite list if "k" = 3.

harvtxt|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 harvtxt|Berge|1989.

Line graphs of "k"-uniform linear hypergraphs, "k" ≥ 3

A global characterization of Krausz type for the line graphs of "k"-uniform linear hypergraphs for any "k" ≥ 3 was given by harvtxt|Naik|Rao|Shrikhande|Singhi|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. harvtxt|Metelsky|Tyshkevich|1997 and harvtxt|Jacobson|Kézdy|Lehel|1997 improved this bound to 19. At last harvtxt|Skums|Suzdal'|Tyshkevich|2005 reduced this bound to 16. harvtxt|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.

The difficulty in finding a characterization of linear "k"-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for "m" > 0, consider a chain of "m" diamond graphs such that the consecutive diamonds share vertices of degree two. For "k" ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs ofharvs|last1 = Naik|last2 = Rao|last3 = Shrikhande|last4 = Singhi|year = 1980|year2 = 1982|txt = yes as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke's of line graphs of graphs.

There are some interesting characterizations available for line graphs of linear "k"-uniform hypergraphs due to various authors (harvs|last1 = Naik|last2 = Rao|last3 = Shrikhande|last4 = Singhi|year = 1980|year2 = 1982|nb = yes, harvnb|Jacobson|Kézdy|Lehel|1997, harvnb|Metelsky|Tyshkevich|1997, and harvnb|Zverovich|2004) under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least "k"³-2"k"²+1 in harvtxt|Naik|Rao|Shrikhande|Singhi|1980 is reduced to 2"k"²-3"k"+1 in harvtxt|Jacobson|Kézdy|Lehel|1997 and harvtxt|Zverovich|2004 to characterize line graphs of "k"-uniform linear hypergraphs for any "k" ≥ 3.

The complexity of recognizing line graphs of linear "k"-uniform hypergraphs without any constraint on minimum degree (or minimum edge-degree) is not known. For "k" = 3 and minimum degree at least 19, recognition is possible in polynomial time (harvnb|Jacobson|Kézdy|Lehel|1997 and harvnb|Metelsky|Tyshkevich|1997). harvtxt|Skums|Suzdal'|Tyshkevich|2005 reduced the minimum degree to 10.

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

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