Common partial transversals and integral matrices

Author:
R. A. Brualdi

Journal:
Trans. Amer. Math. Soc. **155** (1971), 475-492

MSC:
Primary 05B40; Secondary 05A05

MathSciNet review:
0313093

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Abstract: Certain packing and covering problems associated with the common partial transversals of two families and of subsets of a set are investigated. Under suitable finitary restrictions, necessary and sufficient conditions are obtained for there to exist pairwise disjoint sets where each is a partial transversal of with defect at most and a partial transversal of with defect at most . We also prove that (i) where each is a common partial transversal of and if and only if (ii) where each is a partial transversal of and (iii) where each is a partial transversal of . We then derive necessary and sufficient conditions for the validity of (i).

The proofs are accomplished by establishing a connection with these common partial transversal problems and representations of integral matrices (not necessarily finite or countably infinite) as sums of subpermutation matrices and then using known results about the existence of a single common partial transversal of two families. Accordingly various representation theorems for integral matrices are derived.

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0313093-3

Keywords:
Packing problem,
covering problem,
partial transversal,
transversal,
defect of a partial transversal,
common partial transversal,
integral matrix,
subpermutation matrix,
row and column defect of a subpermutation matrix,
permutation matrix,
representations of an integral matrix,
linking principle

Article copyright:
© Copyright 1971
American Mathematical Society