Splittings of Toric Ideals

Let $I\subseteq R=\mathbb{K}[x_1,\dots ,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be “split” into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I=I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I=I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.