Algebraic properties of toric rings of graphs

Betti numbers of toric ideals of graphs: A case study

“…The number β i,j (I G ) is called the i, j-th graded Betti number of I G and equals the number of minimal generators of degree j in the i-th syzygy module of I G . Ideally, we would like to determine β i,j (I G ) directly from a description of G. A non-exhaustive list of papers that have worked towards a dictionary between the invariants of the minimal resolution and G includes [1,2,7,8,10,13,14,16,18,20,24]. However, given that it can be difficult to enumerate the minimal generators of I G , i.e., the numbers β 0,j (I G ), a general solution to this problem is currently unknown.…”

“…Note that our special family can give us some partial insight into the graded Betti numbers of I G for more general graphs. In particular, if G r,d is an induced subgraph of G, we can apply work of Beyarslan, Hà, and O'Keefe [1] to show β i,j (I G r,d ) ≤ β i,j (I G ); Theorem 4.1 uses this inequality to give lower bounds on the Castelnuovo-Mumford regularity and projective dimension for any toric ideal I G .…”