Non-vanishingness of Betti numbers of edge ideals

Abstract

Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.

12 pages. v2: Since Corollary 5.6 was proved by S. Morey and R. H. Villarreal, we added the reference; v3: We changed the explanation before Corollary 5.6, added a reference, and corrected typos

References in corpus (1)

• Betti Numbers of Graph Ideals

Cited by in corpus (2)

• Projective Dimension, Graph Domination Parameters, and Independence Complex Homology
• Regularity and projective dimension of edge ideal of $C_5$-free vertex decomposable graphs