Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity

TL;DR

This paper investigates bipartite graphs with sequentially Cohen-Macaulay edge ideals, establishing that their independence complexes are vertex decomposable and providing a way to compute their regularity from graph invariants.

Contribution

It proves that such bipartite graphs have vertex decomposable independence complexes and offers a method to determine their regularity from graph invariants.

Findings

Independence complex of G is vertex decomposable.

Regularity of R/I(G) can be computed from G's invariants.

Characterization of bipartite graphs with sequentially Cohen-Macaulay edge ideals.

Abstract

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.