A palindromicity criterion for the $h$-polynomials of bipartite edge rings

Abstract

We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_{1}t+\cdots +h_s{t}^s$. We prove that if $\mathbb{k}[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\mathbb{k}[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\mathbb{k}[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise minimal Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial.

12 pages