Showing papers by "Richard A. Brualdi published in 1969"

Comments on bases in dependence structures

Abstract: Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B1 and B2 are bases of a dependence structure, then there is an injection σ: B1 → B2 such that (B2 / {σ(e)}) ∩ {e} is a basis for all e in B1. A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B1 and B2 are bases of a dependence structure then given e in B1 there is an f in B2 such that both (B1 / {e}) ∩ {f} and (B2 / {f}) ∩ {e} are bases. This is a symmetrical kind of replacement theorem.

A very general theorem on systems of distinct representatives

Abstract: 1 Introduction Currently the theory of systems of distinct representatives (and the closely allied theory of transversals) is being carefully examined and reworked, often in a more general context which allows for the transfinite situation This theory can be said to have had its beginning in 1935 when P Hall proved his now celebrated theorem for the existence of a system of distinct representatives of a finite family of sets In a no less significant paper M Hall, Jr (in 1948) extended P Hall's theorem to infinite families of finite sets Around these two theorems a considerable literature has grown (for an excellent survey and thorough bibliography see [10]) The two theorems have been refined in various ways by requiring that the system of distinct representatives have additional properties It is however true that these refinements can be obtained by applying the original theorems to a modified family of sets For finite families this is implicit in the work of Ford and Fulkerson [3] who show how most of these refinements can be obtained from their maximum flow-minimum cut theorem for flows in networks For finite or infinite families Mirsky and Perfect [10], [11] have shown how these refinements can be obtained from the original theorems of the two Halls and a generalization of a mapping theorem of Banach [1] In a recent paper [2] we obtained a further generalization of Banach's mapping theorem This theorem along with M Hall's theorem enables us to prove a very general theorem on systems of distinct representatives, which is in fact a transfinite and symmetrized form of a theorem of A J Hoffman and H W Kuhn The theorem we prove contains as special cases (that is, without further refinement) all theorems that we know which assert the existence of a system of distinct representatives of a given family of sets or subfamily thereof with certain properties being required We then can prove a theorem giving necessary and sufficient conditions that a family of sets possess a family of subsets whose cardinalities lie within prescribed bounds and where the frequencies of occurrences in these subsets of the elements lie within prescribed bounds This will be made more precise later From this we also obtain an extension to locally finite graphs of Ore's solution [12] of the so-called " subgraph problem for directed graphs" and for that matter a generalization of Ore's solution due to Ford and

An extension of Banach’s mapping theorem

Abstract: The following mapping theorem of Banach [I] is well known. It is the basis of most proofs of the Schroder-Bernstein equivalence theorem. If X and Y are sets and f: X-* Y and g: Y-*X are injective mappings, then there exists partitions2 X=X1+X2 and Y= Yi+ Y2 such that f (Xj) = Yi and g(Y2) =X2. The conclusion of this theorem can be rephrased in the following way. Let AC XX Y be the relation between X and Y defined by