Bigraphic pairs with an A-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥5 is an odd integer.