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

Abstract

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of 3-disjoint edges in $G$. Moreover for this class of graphs we characterize $\T{pd}(R/I(G))$ and $\T{depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

To appear in: Proc. Amer. Math. Soc

References in corpus (4)

• Vertex decomposable graphs and obstructions to shellability
• Dirac's theorem on chordal graphs and Alexander duality
• Projective Dimension, Graph Domination Parameters, and Independence Complex Homology
• Non-vanishingness of Betti numbers of edge ideals