On the binomial arithmetical rank of toric ideals (Q2874694)

Let \(I\subset K[x_1,\dots,x_n]\) be a toric ideal, that is a prime pure binomial ideal. Important invariants are associated to \(I\). The height \(\mathrm{ht}(I)\), \(\mu(I)\) the minimal number of generators of \(I\), \(\mathrm{ara}(I)\) the minimal number of generators of \(I\) up to radical and \(\mathrm{bar}(I)\) the minimal number of binomial generators of \(I\) up to radical. We have the following inequalities: NEWLINE\[NEWLINE \mathrm{ht}(I)\leq \mathrm{ara}(I)\leq \mathrm{bar}(I)\leq \mu(I).NEWLINE\]NEWLINE So it is natural to ask whether some of them are equalities. In the paper under review, firstly, the author finds a sufficient condition for the equality \(\mathrm{bar}(I)= \mu(I)\). Secondly, the author studies when \(\mathrm{bar}(I)= \mu(I)\) for a toric ideal associated to a graph, and describe a class of graphs for which the equality \(\mathrm{bar}(I)= \mu(I)\) is true.