Frederik Garbe: Infinitely Many Counterexamples to a Conjecture of Lovász скачать в хорошем качестве

Frederik Garbe: Infinitely Many Counterexamples to a Conjecture of Lovász

The matching number ν(G) of a graph G is the maximum number of pairwise disjoint edges. The vertex cover number τ(G) is the minimum cardinality of a set of vertices which intersects every edge. It is a classical result in graph theory, called König's theorem, that τ(G)=ν(G) for every bipartite graph G. For r-partite r-uniform hypergraphs it was conjectured by Ryser that τ(G) ≤ (r-1)ν(G). Moreover, Lovász conjectured in 1975 that one can always reduce the matching number by removing r-1 vertices which would imply Ryser's conjecture. Clow, Haxell, and Mohar very recently disproved this for r=3 using the explicit counterexample of a line hypergraph of a 3-regular graph of order 102. We construct the first infinite family of counterexamples for r=3, the smallest of which is the line hypergraph of a graph of order only 22. In addition, we give the first counterexamples for r=4. This is joint work with Aida Abiad, Xavier Povill, and Christoph Spiegel.