Let \(A\subset {\mathbb N}^n\) be a vector configuration of \(m\) vectors, and \({\mathbb N}A \) the semigroup generated by \(A\). The polynomial ring \(K[x_1,\dots,x_m]\) is graded by \({\mathbb N}A \). The toric ideal \(I_A\) is generated by all the binomials \({\mathbf {x^u}}-{\mathbf {x^v}}\), such that \({\mathbf {x^u}},{\mathbf {x^v}}\) have the same \({\mathbb N}A \)-degree. A binomial \({\mathbf {{x^u}}}-{\mathbf {x^v}}\) is called primitive is there exists no other binomial \({\mathbf {x^w}}-{\mathbf {x^z}}\) in \(I_A\), such that \({\mathbf {x^w}}\) divides \({\mathbf {x^u}}\) and \({\mathbf {x^z}}\) divides \({\mathbf {x^v}}\). The set of all primitive binomials in \(I_A\) is the Graver Basis \(\mathrm{Gr}_A\). An irreducible binomial is called a circuit if it has minimal support.
Let \(t_A\) be the maximal true degree of any circuit in \(I_A\). The true circuit conjecture says that \(\deg(B)\leq t_A\) for any \(B\in \mathrm{Gr}_A\), where \(\deg(B)\) is the usual degree in \(K[x_1,\dots,x_m]\). There are several families of toric ideals where the true circuit conjecture is true.
The main result of this paper stay that in general, there is no polynomial in \(t_A\) that bounds the degree of every element of the Graver Basis \(\mathrm{Gr}_A\).