Article Submitted to Journal of Symbolic Computation Lexicographic Gröbner bases for transportation problems of format r × 3 × 3 ∗

By means of suitable sequences of graphs, we describe the reduced lexicographic Gröbner basis of the toric ideal associated with the 3−dimensional transportation problem of format r × 3 × 3(r any integer> 1). In particular, we prove that the bases for r =2, 3, 4, 5 determine all others