Line graph of a hypergraph

In graph theory, particularly in the theory of hypergraphs, the line graph of a hypergraph $H$ , denoted $L(H)$ , is the 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 $k \geq 3$ .

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.

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