2018/10/30 by Maria Axenovich, Christian Winter, Axenovich, Maria +1
Mathematics · Engineering · Computer Science · #Limits and Structures in Graph Theory #graph theory and CDMA systems #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.1810.12734
For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all Berge-G hypergraphs is denoted B(G). A hypergraph H is called Berge-G saturated if it does not contain any subhypergraph from B(G), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Each Berge-G saturated hypergraph has at least |E(G)|-1 hyperedges. We show that for each graph G that is not a certain star and for any n at least |V(G)|, there is a Berge-G saturated hypergraph on n vertices and exactly |E(G)|-1 hyperedges. This solves a problem of finding a saturated hypergraph on n vertices with the smallest number of edges exactly.