Extremal Hypergraphs for Ryser’s Conjecture I: Connectedness of Line Graphs of Bipartite Graphs
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چکیده
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. This paper is the first part of the proof of a characterization of those 3-uniform hypergraphs for which Aharoni’s Theorem is tight. It turns out that these hypergraphs are far from being unique; for any given integer k there are infinitely many of them with matching number k. Our proof of this characterization is also based on topological machinery. It turns out that the link graphs of Ryser-extremal 3-uniform hypergraphs are extremal for a natural graph theoretic problem of a topological sort: the independence complexes of their line graphs have minimum (topological) connectedness as a function of their dimension. In this paper we prove a characterization of the bipartite graphs extremal for this problem. This characterization will provide us with valuable structural information for our characterization of Ryser-extremal 3-uniform hyper-
منابع مشابه
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 charact...
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