Addendum To Schrijver's Work On Minimum Permanents

Let ∆n denote the set of n×n matrices of non-negative integers which have each row and column sum equal to k. Let Λn denote the subset of all binary matrices (matrices of zeroes and ones) in ∆n. If G is a bipartite multigraph let B(G) denote the usual ‘biadjacency’ matrix of G. That is, B(G) is the matrix with rows and columns respectively corresponding to the vertices in the two parts of G, and with each entry recording how many edges there are between the vertices corresponding to the row and column in which the entry lies. If G is a k-regular bipartite multigraph on 2n vertices then B(G)∈∆n. Moreover, B(G) ∈ Λn iff G is simple. Schrijver [5] has studied the minimum permanent in ∆n. Our purpose in this note is to use his results to deduce information about the minimum permanent in Λn. As an aside, it has been conjectured by Minc (Conjecture 24 in [4]) that the minimum permanents in Λn and ∆ k n coincide. Our results will be consistent with that conjecture, but do not seem to represent progress towards its resolution.