Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs
The s-line graph \(L_ s\) of a hypergraph \(H=(V,E)\) has E as its vertex- set and \(e_ 1\), \(e_ 2\) are adjacent in \(L_ s\) if \(| e_ 1\cap e_ 2|\geq s.\) First, operations are defined which permit to construct all the hypergraphs with a given 1-line graph starting from the dual hypergraph of this graph. From this: A triangle-free graph is the 1- line graph of a unique hypergraph. As a corollary Whitney's classical theorem on edge isomorphisms of graphs is obtained. A generalization of Whitney's theorem is proved: The strong isomorphisms of two hypergraphs of rank r follows under an additional condition if all corresponding pairs of partial hypergraphs on \(\lceil\log_ 2r\rceil\) of the edges are strongly isomorphic. This implies several corollaries.
scientific article; zbMATH DE number 3873375