An extremal problem on bigraphic pairs with an AA-connected realization

Let S=(a1,…,am;b1,…,bn)S=(a1,…,am;b1,…,bn), where a1,…,ama1,…,am and b1,…,bnb1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn)S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair   if there is a simple bipartite graph G=(X∪Y,E)G=(X∪Y,E) such that a1,…,ama1,…,am and b1,…,bnb1,…,bn are the degrees of the vertices in XX and YY, respectively. Let AA be an (additive) Abelian group. We define σ(A,m,n)σ(A,m,n) to be the minimum integer kk such that every bigraphic pair S=(a1,…,am;b1,…,bn)S=(a1,…,am;b1,…,bn) with am,bn≥2am,bn≥2 and σ(S)=a1+⋯+am≥kσ(S)=a1+⋯+am≥k has an AA-connected realization. In this paper, we determine the values of σ(Z3,m,m)σ(Z3,m,m) for m≥4m≥4.