Markov complexity of monomial curves

Let $mathcal{A}={{bf a}_1,ldots,{bf a}_n}subsetBbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{mathcal{A}}$. We show that the Markov complexity of $mathcal{A}={n_1,n_2,n_3}$ is equal to two if $I_{mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $rgeq 2$ there is a unique minimal Markov basis of $mathcal{A}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $mathcal{A}$ is greater than $l$.