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).
Read more about Line Graph Of A Hypergraph: Line Graphs of k-uniform Hypergraphs, k ≥ 3, Line Graphs of k-uniform Linear Hypergraphs, k ≥ 3
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