Extremal Hypergraphs for Ryser’s Conjecture: Connectedness of Line Graphs of Bipartite Graphs

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound dim(I(G)) 2 − 2 on the connectedness of the independence complex I(G) of line graphs of bipartite graphs G is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conjecture of Ryser. Ryser’s Conjecture states that any r-partite r-uniform hypergraph has a vertex cover of size at most (r− 1)-times the size of the largest matching. For r = 2, the conjecture is simply König’s Theorem. It has also been proven for r = 3 by Aharoni using a beautiful topological argument. In a separate paper we characterize the extremal examples for the 3-uniform case of Ryser’s Conjecture (i.e., Aharoni’s Theorem), and in particular resolve an old conjecture of Lovász for the case of Ryser-