Bounds on the regularity of toric ideals of graphs

TL;DR

This paper establishes bounds on the Castelnuovo-Mumford regularity of toric ideals of graphs, relating it to graph structure, and provides new insights into Betti numbers for specific bipartite graphs.

Contribution

It introduces new bounds for the regularity of toric ideals based on graph properties and offers a novel proof for Betti numbers of complete bipartite graphs.

Findings

Lower bound for regularity based on induced complete bipartite graphs.

Upper bound for regularity in chordal bipartite graphs.

New proof for Betti numbers of $K_{2,n}$.

Abstract

Let $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in terms of the sizes and number of induced complete bipartite graphs in $G$. When $G$ is a chordal bipartite graph, we find an upper bound for the regularity of $I_G$ in terms of the size of the bipartition of $G$. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph $K_{2,n}$.