Regularity and h-polynomials of Binomial Edge Ideals

Abstract

Let G be a finite simple graph on the vertex set [n] = {1,…, n} and K[x, y] = K[x1,…, xn, y1,…, yn] the polynomial ring in 2n variables over a field K with each \(\deg x_{i} = \deg y_{j} = 1\). The binomial edge ideal of G is the binomial ideal JG ⊂ K[x, y] which is generated by those binomials xiyj − xjyi for which {i, j} is an edge of G. The Hilbert series \(H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )\) of K[x, y]/JG is of the form \(H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )/(1 - \lambda )^{d}\), where \(d = \dim K[\mathbf {x}, \mathbf { y}]/J_{G}\) and where \(h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{0} + h_{1}\lambda + h_{2}\lambda ^{2} + {\cdots } + h_{s}\lambda ^{s}\) with each \(h_{i} \in \mathbb Z\) and with hs≠ 0 is the h-polynomial of K[x, y]/JG. It is known that, when K[x, y]/JG is Cohen–Macaulay, one has \(\operatorname {reg}(K[\mathbf {x}, \mathbf {y}]/J_{G}) = \deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )\), where reg(K[x, y]/JG) is the (Castelnuovo–Mumford) regularity of K[x, y]/JG. In the present paper, given arbitrary integers r and s with 2 ≤ r ≤ s, a finite simple graph G for which reg(K[x, y]/JG) = r and \(\deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = s\) will be constructed.