REGULARITY AND PROJECTIVE DIMENSION OF THE EDGE IDEAL OF C5-FREE VERTEX DECOMPOSABLE GRAPHS

In this paper, we explain the regularity, projective dimension and depth of the 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, 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 pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.