Splittings of toric ideals of graphs

Source abstract

Abstract Let G be a simple graph on the vertex set $$\{v_{1},\ldots ,v_{n}\}$$ { v 1 , … , v n } . An algebraic object attached to G is the toric ideal $$I_G$$ I G . We say that $$I_G$$ I G is subgraph splittable if there exist subgraphs $$G_1$$ G 1 and $$G_2$$ G 2 of G such that $$I_G=I_{G_1}+I_{G_2}$$ I G = I G 1 + I G 2 , where both $$I_{G_1}$$ I G 1 and $$I_{G_2}$$ I G 2 are not equal to $$I_G$$ I G . We show that $$I_G$$ I G is subgraph splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not subgraph splittable. In contrast, we show that the toric ideal of a complete graph $$K_n$$ K n is always subgraph splittable when $$n \ge 4$$ n ≥ 4 . Additionally, we show that the toric ideal of $$K_n$$ K n has a minimal splitting if and only if $$4 \le n \le 5$$ 4 ≤ n ≤ 5 . Finally, we prove that any minimal splitting of $$I_G$$ I G is also a reduced splitting.