Non-vanishing of Betti numbers of edge ideals and complete bipartite subgraphs

TL;DR

This paper investigates conditions under which Betti numbers of edge ideals do not vanish, linking graph substructures to algebraic invariants, and provides a combinatorial formula for the projective dimension of unmixed bipartite graphs.

Contribution

It establishes non-vanishing criteria for Betti numbers based on complete bipartite subgraphs and offers a combinatorial description of the projective dimension for unmixed bipartite graphs.

Findings

Betti numbers do not vanish if the graph contains certain complete bipartite subgraphs

Provides a combinatorial formula for the projective dimension of unmixed bipartite graphs

Links graph substructures to algebraic properties of edge ideals

Abstract

Given a finite simple graph one can associate the edge ideal. In this paper we prove that a graded Betti number of the edge ideal does not vanish if the original graph contains a set of complete bipartite subgraphs with some conditions. Also we give a combinatorial description for the projective dimension of the edge ideals of unmixed bipartite graphs.