Line graphs of hypergraphs I

Abstract We define the k -line graph of a hypergraph H as the graph whose vertices are the edges of H , two vertices being joined if the edges they represent intersect in at least k elements. In this paper we show that for any integer k and any graph G there exists a partial hypergraph H of some complete h -partite hypergraph K h h x N such that G is the k -line graph of H . We also prove that, for any integer p , there exist graphs which are not the ( h - p )-line graph of some h -uniform hypergraph. As a corollary we answer a problem of C. Cook. Further we show that it is not possible to characterize the ( h - 1)-line graphs by excluding a finite number of forbidden induced subgraphs.