The size of Betti tables of edge ideals arising from bipartite graphs

TL;DR

This paper characterizes all possible pairs of projective dimension and regularity of edge ideals for connected bipartite graphs with a fixed number of vertices, providing a complete classification.

Contribution

It provides a complete classification of the pairs (projective dimension, regularity) for edge ideals of connected bipartite graphs on n vertices.

Findings

All pairs (pd, reg) are determined for connected bipartite graphs.

The classification applies to graphs with any fixed number of vertices.

Results facilitate understanding of algebraic invariants of bipartite graph edge ideals.

Abstract

Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs $(\operatorname{pd}(I(G)),\, \operatorname{reg}(I(G)))$ as $G$ ranges over all connected bipartite graphs on $n$ vertices.