Projective Dimension, Graph Domination Parameters, and Independence Complex Homology

Abstract

We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs; in particular we draw heavily from the topic of dominating sets. Through Hochster's Formula, these bounds recover and strengthen existing results on the homological connectivity of graph independence complexes.

Add references for Chapter 5 and 6. Add remarks 5.7 and 5.8

References in corpus (3)

• Vertex decomposable graphs and obstructions to shellability
• Non-vanishingness of Betti numbers of edge ideals
• The word problem and the Aharoni-Berger-Ziv conjecture on the connectivity of independence complexes

Cited by in corpus (1)

• Regularity and projective dimension of edge ideal of $C_5$-free vertex decomposable graphs