Koszulness, Krull dimension, and other properties of graph-related algebras

The algebra of basic covers of a graph G, denoted by (A) over bar (G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of (A) over bar (G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then (A) over bar (G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.