  In this paper we show that if for an integer matrix A the universal Gr\"obner
basis of the associated toric ideal \Ideal_A coincides with the Graver basis of
A, then the Gr\"obner complexity u(A) and the Graver complexity g(A) of its
higher Lawrence liftings agree, too. We conclude that for the matrices
A_{3\times 3} and A_{3\times 4}, defining the 3\times 3 and 3\times 4
transportation problems, we have u(A_{3\times 3})=g(A_{3\times 3})=9 and
u(A_{3\times 4})=g(A_{3\times 4})\geq 27. Moreover, we prove
u(A_{a,b})=g(A_{a,b})=2(a+b)/\gcd(a,b) for positive integers a,b and
A_{a,b}=(\begin{smallmatrix} 1 & 1 & 1 & 1 0 & a & b & a+b \end{smallmatrix}).
