Perfect Matchings in Balanced Hypergraphs – A Combinatorial Approach

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuéjols, Kapoor, and Vušković extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof. Received April 29, 1997     

Fundamentalsätze der metrischen Geometrie

For arbitrary quadratic forms, including the cases of characteristic 2 and of infinite dimensions, several affine-metric and projective-metric structures are considered, and the corresponding isomorphisms are determined. As an application, a general fundamental theorem of the miquelian circle geometry is proved.

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Martin Barner zum 65. Geburtstag gewidmet     

Pairwise sufficiency

Summary For an arbitrary experiment E we investigate the relation between its pairwise sufficient subalgebras and its sufficient sublattices in the M-space of E (in the sense of L. LeCam). By exhibiting an experiment without minimal pairwise sufficient subalgebra it is shown that this correspondence is in general not bijective. In view of this we introduce the rather large class of majorized experiments. They have a minimal pairwise sufficient subalgebra which can be described explicitely.As a natural subclass of the majorized experiments appear the coherent experiments that are distinguished by the coincidence of sufficiency and pairwise sufficiency. It is shown that the coherent experiments are characterized by the fact that they admit a majorizing measure which is localizable. As a consequence we obtain that the class of coherent experiments coincides with classes of experiments previously introduced by T.S. Pitcher, D. Mußmann, M. Hasegawa and M.D. Perlman.