Extremal Betti Numbers of Some Classes of Binomial Edge Ideals

Let G be a simple graph on the vertex set [n] with edge set E(G) and let S be the polynomial ring K[x1, . . . , xn, y1, . . . , yn] in 2n variables endowed with the lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn. The binomial edge ideal JG ⊂ S associated with G is generated by all the binomials fij = xiyj−xjyi with {i, j} ∈ E(G). The binomial edge ideals were introduced in [5] and, independently, in [8]. Meanwhile, many algebraic and homological properties of these ideals have been investigated; see, for instance, [1–3, 5, 7, 9–14]. In [2], the authors conjectured that the extremal Betti numbers of JG and in<(JG) coincide for any graph G. Here, < denotes the lexicographic order in S induced by the natural order of the variables. In this article, we give a positive answer to this conjecture when the graph G is a complete bipartite graph or a cycle. To this aim, we use some results proved in [12] and [14] which completely characterize the resolution of the binomial edge ideal JG when G is a cycle or a complete bipartite graph. In particular, in this case, it follows that JG has a unique extremal Betti number. In the first section we recall all the known facts on the resolutions of binomial edge ideals of the complete bipartite graphs and cycles. In Section 2, we study the initial ideal of JG when G is a bipartite graph or a cycle. We show that proj dim in<(JG) = proj dim JG and reg in<(JG) = reg JG, and, therefore, in<(JG) has a unique extremal Betti number as well. Finally, we show that the extremal Betti number of in<(JG) is equal to that of JG.