Let S = ( a 1 , ? , a m ; b 1 , ? , b n ) , where a 1 , ? , a m and b 1 , ? , b n are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , ? , a m ; b 1 , ? , b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ? Y , E ) such that a 1 , ? , a m and b 1 , ? , b n are the degrees of the vertices in X and Y , respectively. Let A be an (additive) Abelian group. We define ? ( A , m , n ) to be the minimum integer k such that every bigraphic pair S = ( a 1 , ? , a m ; b 1 , ? , b n ) with a m , b n ? 2 and ? ( S ) = a 1 + ? + a m ? k has an A -connected realization. In this paper, we determine the values of ? ( Z 3 , m , m ) for m ? 4 .