Intersection Graphs of k-uniform Linear Hypergraphs

On line graphs of linear 3-uniform hypergraphs

“…We show that a similar characterization is impossible for graphs in L l r with vertex degrees at least c if r > 3 and c is an arbitrary constant (Theorem 3.3). In fact, this was conjectured in [7]. Nevertheless, such a finite characterization was obtained in [7] with a restriction on edge degrees.…”

“…Lovász [6] posed the problem of characterizing the class L 3 and noted that it cannot be characterized by a finite list of forbidden induced subgraphs (see also [3]). Such a characterization is also impossible for the class L l 3 [7]. A global characterization of Krausz type for the class L r for arbitrary r was given by Berge [2] (we call such characterizations Krausz characterizations).…”

“…A global characterization of Krausz type for the class L r for arbitrary r was given by Berge [2] (we call such characterizations Krausz characterizations). A Krausz characterization of the class L l r is obtained in [7]. It is pointed out in [2] that any graph is line for some linear hypergraph.…”

“…Unfortunately, we do not know the complexity of the analogous problem for the class L l r . This paper is inspired by the paper [7] giving a finite induced subgraph characterization of graphs in L l 3 whose vertex degrees are at least 69. In Section 2 of our paper, this degree bound is reduced to 19, and the list of subgraphs differs from the list in [7].…”

“…This paper is inspired by the paper [7] giving a finite induced subgraph characterization of graphs in L l 3 whose vertex degrees are at least 69. In Section 2 of our paper, this degree bound is reduced to 19, and the list of subgraphs differs from the list in [7]. The proof is based on the theorems of Krausz [4] and Beineke [1], the Krausz characterization of the class L l 3 [7] and properties of graph cliques from this class obtained in [5].…”