Line hypergraphs

Abstract

In this paper, we introduce a new multivalued function L called the line hypergraph. The function L generalizes two classical concepts at once, namely, of the line graph and the dual hypergraph. In terms of this function, proofs of some known theorems on line graphs can be unified and their more general versions can be obtained. Three such theorems are considered here, namely, the Berge theorem describing all hypergraphs with a given line graph G in terms of clique coverings of G (Berge, 1973, p. 400), the Krausz global characterization of line graphs for simple graphs (Krausz, 1943) and the Whitney theorem on isomorphisms of line graphs (Whitney, 1932).