Intersection Graphs of k-uniform Linear Hypergraphs

Abstract

A finite hypergraph H is said to be linear if every pair of distinct vértices of H is in at most one edge of H. A 2-uniform linear hypergraph is called a graph. The edge-degree of an edge of a graph G is the number of triangles in G containing the given edge. In this paper it is proved that there is a finite family F of graphs such that any graph G with minimum degree at least 69 is the intersection graph of a 3-uniform linear hypergraph if and only if G has no induced subgraph isomorphic to a member of F . Further, it is shown that there is a polynomial f( k ) of degree less than or equal to 3 with the property that given any integer k (⩾ 2) there exists a finite family F ( k ) of graphs such that any graph G with minimum edge-degree at least f( k ) is the intersection graph of a k -uniform linear hypergraph if and only if G has no induced subgraph isomorphic to a member of F ( k ) .