Markov complexity of monomial curves

Let A={a1,…,an}⊂Nm. We give an algebraic characterization of the universal Markov basis of the toric ideal IA. We show that the Markov complexity of A={n1,n2,n3} is equal to 2 if IA is complete intersection and equal to 3 otherwise, answering a question posed by Santos and Sturmfels. We prove that for any r≥2 there is a unique minimal Markov basis of A(r). Moreover, we prove that for any integer l there exist integers n1, n2, n3 such that the Graver complexity of A is greater than l.