Betti numbers of normal edge rings (II)
Zexin Wang, Dancheng Lu
TL;DR
This work computes the Betti numbers of edge rings for multi-path graphs by applying the induced-subgraph approach to the toric edges ring framework. It classifies multi-path graphs into even, odd, and mixed types, showing that mixed types factor as tensor products of their even and odd parts, and then derives explicit multi-graded Betti-number formulas from induced subgraphs. For even-type graphs, the Betti numbers are captured by sets $E_i^{j}$; for odd-type graphs, by sets $O_i^{j}$; and for mixed types, by combining the contributions from the even and odd parts, along with an exact expression for the projective dimension and regularity. The results express the top-Betti data $\mathcal{N}_G$ in terms of graph parameters, providing complete graded Betti-number profiles and enabling direct computation from the graph structure, with the square-free initial-ideal framework aligning extremal Betti numbers as shown by Conca–Varbaro.
Abstract
We compute the Betti numbers of the edge rings of multi-path graphs using the \emph{induced-subgraph approach} introduced in \cite{WL1}. Here, a multi-path graph refers to a simple graph composed of several paths that have the same starting point and the same ending point. Special cases include the graph $G_{r,d}$ introduced in \cite{GHK}, the graph $G_{r,s,d}$ introduced in \cite{NN}, and the graph $B_{\underline{\ell},h}$ introduced in \cite{LZ}. In particular, we show that all the multi-graded Betti numbers of a multi-path graph are the top multi-graded Betti numbers of some of its induced subgraphs.