Common partial transversals and integral matrices

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.