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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Common partial transversals and integral matrices
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by R. A. Brualdi PDF
Trans. Amer. Math. Soc. 155 (1971), 475-492 Request permission

Abstract:

Certain packing and covering problems associated with the common partial transversals of two families $\mathfrak {A}$ and $\mathfrak {B}$ of subsets of a set $E$ are investigated. Under suitable finitary restrictions, necessary and sufficient conditions are obtained for there to exist pairwise disjoint sets ${F_1}, \ldots ,{F_t}$ where each ${F_i}$ is a partial transversal of $\mathfrak {A}$ with defect at most $p$ and a partial transversal of $\mathfrak {B}$ with defect at most $q$. We also prove that (i) $E = \cup _{i = 1}^t{T_i}$ where each ${T_i}$ is a common partial transversal of $\mathfrak {A}$ and $\mathfrak {B}$ if and only if (ii) $E = \cup _{i = 1}^t{T_i}’$ where each ${T_i}’$ is a partial transversal of $\mathfrak {A}$ and (iii) $E = \cup _{i = 1}^t{T_i}''$ where each ${T_i}''$ is a partial transversal of $\mathfrak {B}$. 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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 155 (1971), 475-492
  • MSC: Primary 05B40; Secondary 05A05
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0313093-3
  • MathSciNet review: 0313093